First, with the top up and secured, check the hinge pins for wear. Open one door at a time just so it clears the body and pick up on the very back edge of the door. If you can lift the door any appreciable amount without lifting the side of the car as well, you need new hinge pins and bushings. You should be able to feel the slop int he hinges, generally, the driver's side will be much worse than the passenger.
The true test of integrity on a convertible is to drop the top and open booth doors. If you cannot close the doors as easily as you did with the top up, you need to investigate the structure of the car before you do anything else. If your quarters are rotted badly, there's a pretty good chance the underside of the car needs help as well. I once looked at a convertible that we had to jack up under the rockers to close the doors....
