Personally...
I have an XBox 360 and a Wii, as well as a PS2 (who doesn't own a PS2?). When I'm at home, I play them all, and I enjoy it. However, I'm at school right now, and in a tiny dorm room. There's no room for a TV, even a small one, and console gaming systems.
Enter the handhelds. They're perfect for the college environment, and wherever and whenever else you happen to be not either at home or actively engaged in something else. With the DS and the PSP, it is extremely easy to put down a game at a moment's notice, and pick it up later, right where you left off. They may lack the sparkly graphics or huge games of the consoles, but, for me at least, their convenience more than makes up for it.
Many of the games for the PSP and the DS are just remakes of older console games. Some may take this as a sign that portable systems are dying; they aren't getting good, NEW content. I argue that this is simply good business practice. Portables can't compete with consoles for graphics or sheer quantity of game-play, because they lack the hardware. However, they CAN compete with the consoles of last generation. It's a sure money-maker to re-release games for previous consoles in portable form, and that is happening more and more frequently.
Personally, I couldn't be happier; this gives me the chance to pick up games I missed the first time around (such as Chrono Trigger, Fire Emblem, and Final Fantasy Tactics), and gives me more access to games I already know I love (such as Disgaea 1 and 2). I don't yet have the recently released Disgaea 2 for the PSP, but I will as soon as I can find $30. Which is another point in favor of handhelds; the games tend to be at least $20 cheaper.