Use a hygrometer to determine relative humidity. Excessive humidity is normal in many basements in summer. Depends on the climate.

However, too many people figure if they see a dehumidifier it means there's something wrong with the home. So I've learned to hide ours away when prospective buyers are taking the tour.

Is there anything we should look for?

I've bought dozens of homes. Signs of water penetration and hidden damage are the biggest concerns. You've already learned to use your nose!

Structural integrity is the next thing. Look for unusual settling, cracks (including patched ones) in corners, and repairs--added support posts, for instance.

If the exterior is barrier EIFS (synthetic stucco with no weeps), get a professional inspection. Otherwise, walk around and look carefully at the condition of the trim, drainage and settling.

You should take a look in the attic--just pop your head in and shine a light to check for waterstains, indicating leaks on the underside of the decking.

I've usually bought homes built from the 1880s to the mid 1950s. So I'm not looking for perfection, but for fatal defects.

Here is a link that might be useful: Humidity Control

A humidifier or lack there of does not tell you anything really. Like Worthy said I would look for signs of water intrusion. A basement that does not leak can have humidity issues too. In the midwest where I live a basement in summer is pretty much garanteed to have high humidity - warm moist summer air moved into a cooler basement will increase the relative humidity. I run a humidifier in our basement just to control humidity due to the lower temps, not because of water leakage.

Thanks for the input. Wouldn't you know it, another buyer beat us to the punch and the seller accepted their offer before even seeing ours. Soooo, we decided to built a new home. It will take a few months (or more)to complete and THEN I am sure there will a million questions on finishing the basemnt. The the meantime, thanks again for the help.