The primary check on the President's ability to wage war is the purse controlled by Congress: that's why the legislature ultimately has control. However, the President is Commander and Chief under the Constitution, so if Congress allocates $800 billion for defense, he can do a lot of commanding and chiefing without needing to ask for more money. You can't have it both ways: if the world is so dangerous that we need $800 billion defense budgets, then Congress can't exercise command over the use of the forces it has paid for. That's by Constitutional design.
If you read the Constitution, the primary concern of the Declaration of War powers allocated to Congress is American liberty: only after a declaration by Congress can the President charge someone with treason. The Founders didn't give a rat's patooty about foreigners-- they had their own governments and powers in the 18th Century made and broke treaties all the time and skirmished all the time.
Last, we still have what SCOTUS has already deemed a Declaration of War in place: the AUMF following 9/11. The Houthis have attacked US ships, so I don't think a declaration is needed to retaliate. If it is, we already have one because the Houthis are attacking to defend Hamas they're classified as terrorists under the AUMF.
The problem isn't the attack on the Houthis. The problem is that Biden was erected not elected President and the Uniparty Congress is more interested in its own graft than in exercising oversight.