I am sure we probably aren't doing everything we can to deal with the influx. At minimum we could at least mobilize a mass humanitarian aid project like we did for the Palestinians, we didn't need Congressional Action to do that. Deploy the Red Cross to the local air bases along the border. Show these people getting treat humanely while Texas wants to do nothing short of shooting them or letting them drown.
But the Dems inaction is outshines the callousness of the Republicans yet again.
I think the Democrats are stuck between withering criticism/obstruction by the center to the far right and then not enough help from the left leaving them no political room to do much.
But remember, CONGRESS is charged with creating laws on immigration, which they have done very little on since Reagan due to atrocious politics of othering the Southern immigrants. The far right only wants to kick out every non-citizen and even some USA citizens they don't like, so until the far right are put back in their Pandora's box, they will spoil every attempt at creating a functional immigration system based in reality.
It will take a complete crisis inside of the country to improve our immigration system, and there's nothing the right or left, DNC or RNC can or will do to change much at the border except:
We could stem the crises by a new Marshall Plan to make the countries more desirable for the immigrants to stay in. But I'm a dreamer, and I doubt that will ever happen.

