Well, first of all, you may want to reconsider the top you've proposed. How are you joining the boards in the miters, the outer border, and the field planks? Mortises, spines, biscuits? You might want to consider a European cabinet design (5-sided box)and put a drawer in it. Shear support is most important to prevent collapse due to weight and lateral forces of the machines.

To answer your initial question, gussets are typical for mitigating shear.

Well, that's just the top look I wanted. Even though this will be under the washer & dryer, I wanted it to look decent.

I wasn't going to join the planks as much as just screw them into underlying supports. I just am not sure how to setup those beams to support the weight and vibrations.

First of all, any built up for a washer/dryer has to have a rim around the top to prevent the machine from moving and falling off.

Second, building it high enough to store laundry baskets should create sufficient height to need a lot of bracing, which will probably take up the 'storage' space.

If you look at laundry room design with built up areas for the washer/dryer, you will see almost all are permanently built into the wall/floor.

There is not much way you can successfully design and build that type of riser with wood.

Build a welded steel tubing frame from square stock and clad it with wood any way you want.
You can't get an open front any other way. An open fronted wood stand will lack rack bracing needed for the very heavy and vibration-prone washer.
Casey