There are three different aspects to the question: link layer safety, Internet protocol safety, and exposure risk. Let me take them each in turn.
First -- once packets are "in the Internet", they're all equal, no matter the source. Different ISPs can be better or worse on security, but that does not depend on how the packets reach the Internet. Broadband is mostly about what has been called "the last mile", the link from an ISP's POP (point of presence) to the consumer. Modems are the traditional technique, but they're slow, so we're seeing deployment of broadband links. Most of the security issues focus on these links.
It's hard to tap traditional modems, though this can be and has been done by, say, law enforcement. (There was an article in the 1 October 1999 Wall Street Journal on a hacker case in Texas, where the FBI had to have special equipment built.) Nevertheless, it is difficult, and is not a threat for most people.
Some broadband technologies, such as satellite systems and cable modems, present more of a risk in this regard, because the packets traverse a shared medium. (DSL, on the other hand, uses a dedicated cable pair, just as a traditional telephone line does.) It is indeed possible to eavesdrop on some cable modem systems, especially the older ones. For technical reasons, it is generally much easier to listen to downstream (i.e., to the user) communications than upstream; this in itself is some protection, because the most sensitive data (passwords, credit card numbers, and the like) travels upstream only. Still, even downstream data should be protected from third parties; accordingly, the new cable modem standards (the DOCSIS standards) do, in fact, require encryption, for precisely this reason. (The Cablelabs article on cable modem security, is in my opinion, quite fair and accurate.)
If the cable modem doesn't do encryption (some older ones don't), it may or may not be possible to eavesdrop easily. One could always listen to the RF signal directly, but that is, in fact, almost equivalent to building one's own cable modem. The cable modem may filter out IP addresses not assigned to you, which would block simple software spying; the would-be eavesdropper would have to hack the cable modem, either physically or over the net, to disable or modify the filters. This can be difficult but is not impossible. Bottom line: DOCSIS-based cable systems are immune to eavesdropping. On some older systems, it's easy; on others, it's hard. (One caveat: DOCSIS specifies DES encryption. This is not secure against a serious enemy, but it's likely sufficient for cable link security; since the encryption terminates at the cable head end in any event, serious enemies will eavesdrop past that point.)
But there's one important caveat: the existence of encryption doesn't mean that it's being used. Some cable operators do not enable encryption, even if it's available. Check with your operator to be sure.
I'm not familiar with the details of satellite-based systems; the basic issues are similar to those for cable, but there's one added complication: satellite systems are generally asymmetric, and require a telephone line for upstream traffic. (Some cable modems work that way, too.) There is somewhat more risk for trouble here, since the telephone-based upstream link is not closely coupled to the downstream link. This can (note -- "can", not "will") make it more difficult to prevent IP spoofing attacks. But those are a risk to the rest of the Internet, not to the user, and whether or not this in fact a problem would very much depend on the details.
The second question is Internet protocol safety. This is the point most often cited by those who criticize the security of always-on systems. For example, attackers can (and do!) scan cable-modem connected machines for exposed services. Of course, those same attackers can also scan the address ranges assigned to dial-up ports. If you're running a vulnerable service (critics often cite the Microsoft file-sharing services; I've also seen attackers looking for well-known holes in Linux systems), you're at risk, whether you're connected via a broadband link or via a dial-up. Link-layer encryption does nothing to protect you here; however, home firewall products or (some) corporate encrypting tunnels can protect you. For example, you may have an IPsec tunnel to your corporate net; the tunnel product (there are both hardware and software versions available) can and should drop unencrypted, unauthenticated packets.
At one point, there was a rumor that some versions of Windows (probably Windows '95, and not '98) would automatically turn on file-sharing if you installed a network card. That would present an additional risk, not because of the fact of broadband access, but because you're doing something different. I've never been able to confirm this story; it is true, though, that many people have had home or small business nets that they use to share files, not realizing that this would expose them once they connected some or all of those machines to cable modems. Some cable ISPs strongly recommend that you disable file-sharing.
Newer versions of Windows run more services, and thus are inherently more vulnerable to attack. The number of bugs out-of-the-box Windows systems is frightening. You must keep up to date on patches -- and even that isn't enough, because sometimes, the attack rate is so high that a new machine will be penetrated while the owner is trying to install the patches.
There is another big difference from dialup. With almost all dialup systems, you get a new IP address every time you connect. Most (but not all) cable modem and DSL systems assign you a permanent IP address. That means that if your system is, in fact, vulnerable, an attacker can find you again and again, and can keep coming back for more. You're harder to find again if you dial up, or if your broadband provider uses dynamic address assignment. On the other hand, using a constant address makes life easier for you under certain circumstances. I sometimes contact my home machines from my office, because I know what addresses they'll be on.
There is one other danger peculiar to broadband that has developed in recent years. The owner of a broadband-connected machine has a resource that some attackers want: the broadband connection itself. Such a machine is useful for a variety of nefarious purposes, including attacking other machines and launching spam. You do not want your connection used this way; if nothing else, it will slow you down. Besides, your ISP is likely to pull the plug on any such offending machine.
In other words, being on a broadband network makes you more of a target. The bad guys know the IP address ranges associated with broadband ISPs, and scan them frequently. You may not be more vulnerable to an attack, but you are at more risk.
Bottom line: while the biggest danger is from the services you run, you need to be more careful with a broadband connection. It doesn't mean you shouldn't have one; it does mean you need to be more careful.
Another change is being connected all the time. Obviously, if you use a service more, you're more exposed; dashing across the road is not the same as dancing non-stop on the median line. But the biggest risks to most home users are from their Web browsers and their mailers, especially given all the active content on the net.
Bottom line: the more you use a service, the more you're exposed to any risks of it. But the added risks come from using the service more, not how you get to it.
But -- and this is important -- broadband may let you do things that you couldn't easily do before, and this may increase the risk. Again, though, the change comes from doing new things, not from the means of access. Napster is a lovely example. Apart from questions of copyright law, most people can't effectively share MP3s from their dial-up machines, because the connection is too slow. If someone tries to upload a song from you, you can't easily use your dialup connection for anything else; the link is too crowded. That has the consequence of implicitly discouraging use of the service, since it hurts you to use it. At the same time, of course, that protects you. Gnutella is a more interesting case, since it lets you share arbitrary files, and not just MP3s. If it's buggy -- and bugs in it wouldn't surprise me in the slightest -- you're at great risk.
Bottom line: two hundred years ago, no one received crank phone calls or wrong numbers at 3am. But I don't think that most people would want to give up telephones entirely, just because there are some down sides to the technology.