I'd say Illinois is the clear #1 with their all-time win percentage and 5 Final Fours. OU is next.
After that it's hard to rank. Notre Dame has one of the better win percentages and a boatload of Sweet 16 appearances, but they only have a single FF. Texas has a surprisingly high all-time win percentage, a lot of tourney appearances, and 3 FFs. Purdue has been very solid, but they only have 4 EEs and 2 FFs. WVU has a solid all-time record but only 3 EEs and 2 FFs. UVA has 2 FFs but a weaker all-time record. LSU has 4 FFs and an even weaker all-time record. GTech's all-time record is bad. UH has 5 FFs but has been a nobody for more than 30 years. Mizzou has been solid, but I can't rank a program that has both zero FFs and an all-time win percentage outside of the top 50 over any of these others.
I'd rank the top 5 like this:
1. Illinois
2. OU
3. Texas
4. ND (could be #3 instead, if you can overlook the lack of FFs)
5. Purdue