It totally depends on the paper you are using. What paper is giving you this problem?

I find that Arches Platine, Rives BFK, Whatman's Watercolor, Somerset Satin and Fabriano Uno all are VERY beefy papers that can stand up to a lot of wet processing without damage.

Cranes platinotype, Socorro, Lenox and Bienfang 360 are a little more flimsy, but can usually be used without too much extra care.

Masa in my experience is at the bottom of the heap when it comes to wet strength. I never even change trays when I print on that paper, and just pour the chemicals in and out so I can avoid touching it, and then float it out of the rinse water directly onto the drying screen.

So, in short, it depends...