Test for moisture by taping 2-foot squares of plastic sheeting to the floor and walls. Wait a couple of weeks. If condensation forms underneath, your foundation's not sealed. If droplets form on top, your basement needs dehumidifying.
Use an awl to probe for rot and insect damage in floor joists, rim and header joists, the sill plate, and wood-framed windows.
Check floor joists for sagging by climbing a ladder until you're nearly eye level with the underside of the joists. Look across them (perpendicular to their direction) to see if any are out of line.
Have a pro check fuel-burning equipment and your house's ventilation system to ensure that you won't have carbon monoxide buildup below grade.