Issues with the Original
The original PSP had problems with the buttons--the square button was too close to the screen, and could damage it. The memory cover was easily torn off, and the Universal Media Disk (UMD) mechanism was prone to failure.
PSP-2000
The PSP-2000 was about a third the size and weight of the original PSP. The square button issue was fixed. A metal frame that supported the original was removed, and created more flex in the system, leading to cracks in the plastic.
PSP-3000
The PSP-3000 had a number of improvements over the previous models. The UMD drive was redesigned to have fewer moving parts. The screen has improved contrast ratio and pixel response time. The memory cover was redesigned, and the analog nub was made more flexible and hard to remove accidentally.
Remaining Issues
The issues that plagued the original PSP have vanished, but the build quality problems have been an issue with both the PSP-2000 and PSP-3000. The lack of a metal frame is probably the largest issue, leading to cracks in the plastic body of the system and the battery cover.
Considerations
Overall, the PSP-3000 is still probably the best system to get. It has the smallest number of problems of any of the traditional PSP systems. The PSP Go is also a consideration, though not for those who have large UMD libraries, as the Go has no UMD drive. Also, the PSP-3000 is the latest model that supports modding, and the PSP Go has not yet been cracked.