You could price it accordingly, and just sell it as is. Waterproofing can be very expensive.

Good luck!

Yes, damp basements can be a very expensive problem to correct. Really, the waterproofing should be on the outside of the stem walls, and the drainage tiles should be at the base of the foundation. Obviously, it would take a lot of excavation to do it right. A sump pump will get the puddles off the floor as long as the floor slopes towards the sump pit properly, but the moisture still remains due to the water in the pit, and will contribute to a musty smell. Waterproofing the inside of the basement walls, will only provide limited results because the moisture often comes through joints between the walls and the floor, and the water still gets into the wall but not into the room. Really, if you're gonna do it, do it right, which is expensive, so disclose it and sell it the way it is. Damp basements are not uncommon. Some other "fixes" include dehumidifiers and ventilation, which are not very expensive, but require mantainence.

I'm thinking that this is more than a damp basement but one that actually gets water if you had a contractor suggest french drains and a sump pump to the owner. If no one has seen actual water in the basement, a dehumidifer and things like baking soda may work wonders to absorb orders and help to keep it dry. The contractor you called I gather is an expert in waterproofing basements; an important distinction, if not, get one who is and who comes recommended. I've used Bonded myself for homes with water problems with great results according to the owners. If the contractor gave you a written estimate to reverse the problem, I would file it in the file so that when you do get a buyer you can disclose the problem and give them an estimate for the repair which they can take into consideration with their offer. The seller needs to know that this issue will have to be disclosed.
The mold issue is something else. Mold should be disclosed as well. If it is bad then you may need to get a company that does mold remediation. Also something that is not inexpensive to do.

As far as being afraid to come across as pushy, in today's market especially, many sellers not only want but often need to sell their homes. You do a disservice to them if you aren't honest about what needs to be done in order to get the results they want. It's not what is said but how it is said to the client. Don't insult your client's home but let him know your professional opinion; after all isn't that why he hired you?