37 mins for a UA to UX connection in DEN is very achievable - especially if your brother only has hand luggage. I do these types of connections all the time and have never missed a flight.

Yes, the B terminal is long but it's all continuous and does not require any trains or buses. A fit person can easily walk/jog the entire length in 5-10 mins.

Other point is UA is notorious for generously padding their schedules and flights often arrive 10-15 min early. Weather should not be an issue.

If your brother is sitting anywhere near the front and can quickly deplane and get moving to the connecting gate then it's very possible.