That depends on how you define "welfare."

This article looks at food assistance:

Of these, about one-in-five (22%) of Democrats say they had received food stamps compared with 10% of Republicans. About 17% of political independents say they have received food stamps.

This article looks at entitlements generally:

While the two parties are sharply divided over entitlement spending, the differences in the proportions of Republicans and Democrats who have received entitlements is fairly modest: 60% of Democrats, 52% of Republicans and 53% of independents have benefited from one of these six major classes of federal entitlement programs.

So at least from those two studies, Democrats use welfare more than Republicans. The "red states get more welfare funding" notion can be understood to mean that the poor in those states probably vote Democrat and use lots of federal welfare.

Here's another article from another source:

Hardly surprising, we see that in a two-party split, 60-80% of welfare recipients are Democrats, while full time Workers are evenly divided between parties.