The US was part of a UN Police Action in Korea (voted on when the Soviets boycotted the Security Council meeting and thus lost an opportunity to veto) — it was not just the US itself intervening unilaterally. For the Soviets to directly intervene after that point would mean having Soviet troops fighting US troops, which would have been extremely dangerous and carried with it huge risks for escalation. Instead, the Chinese (then still a non-nuclear power) intervened in that way (at not inconsiderable risk to themselves).

But separately, Stalin was never entirely keen on the Korean War. This is something we only learned once the Cold War ended. Kim Il Sung pushed Stalin to endorse his effort and to give help; Stalin feared it was "adventurism" (a risky move) at a point where the Soviets were trying to posture big but in reality were quite vulnerable (the Soviets had only just gotten nuclear arms, but they were very limited in number and had no means of delivering them against the USA, whereas the US had hundreds of nuclear weapons at that point and foreign bomber bases within striking distance of Moscow). Stalin was willing to give some assistance but was not interested in risking anything that might force the US to have a direct confrontation with the USSR.

On the Stalin/Kim discussions, John Gaddis talks about this in The Cold War: A New History, if I recall correctly, and how post-Cold War revelations have shaped our view of this (it was originally thought that Stalin was very keen on the Korean war; archival documentation shows he was not).