There is nothing the House can do unilaterally other than reject the bad bills coming out of the Senate.
All bills related to funding are the primary purview of the House, regardless of whether something budget/money wise originates in the Senate or not (in the end a Senate money bill has to get approved in a joint reconciliation meeting).
Deny the money, deny the action. Or at least make the Democrats use money they wanted to pi$$ away on something else.
I get tired of hearing that excuse. The House controls appropriations, and the power to DENY funding for the Left’s agenda is a powerful club to wield. They can and should be willing to stop the entire government if necessary to get dems and RINOs to actually live up to their oaths and put the country’s security before their own craven personal interests. It is beyond belief that we actually have American politicians refusing to secure the country’s border unless they first get support for their personal political agenda. They should be thankful that we’re only threatening funding for their pet agenda instead of immediately hanging them from lampposts for such treason.