Well, there might be a couple of issues here. But I don't think it is all that uncommon to have two dishwashers.
Of course, the most important thing is going to be how you connect the drain hoses. If you have a garbage disposal with a drain connection for a dishwasher, it's only going to accommodate one dishwasher. So you could connect one to the disposal, and connect the other into the tailpiece under the sink basin opposite the disposal. You have to put a high under-sink loop in both drain hoses at the very least. Check your local code, though. There's a good chance you have to install an air-gap device on each drain hose, and you might have to mount the air gaps above the counter top.
So the issue you might be reading about is backflow, which is when dirty water flows backwards into your dishwasher. Generally, the high loop(s) and, if required, the air gap device(s) will prevent this.
Another related issue is that it's important to drain the dishwasher(s) into the sewer line before the P trap. The water plug sitting in the P trap (combined with proper venting) is what keeps sewer gases from pushing back up into your house. If you connect the dishwasher drain hose(s) downstream of the P trap, you subvert this mechanism and sewer gases will push up into your dishwasher.
Back pressure might be a different matter, if the drain pipe under your sink is too small to handle both dishwashers draining at the same time, or if the pipe is clogged, then the water pressure could be high enough to force some of the water back up the drain pipe into your sink.
having just redone the dishwasher, sink, disposal and faucet/drains in my kitchen I can hopefully help you here. You shouldn't need an airgap for one. I actually removed mine when I redid them because I found it to be ugly and both the dishwasher and disposal manuals say one was not necessary. What you can do is loop the dishwasher drain hose so it reaches above the connection to the disposal, this should keep stuff from running backwards into the dishwasher. I am however concerned that this is happening in the first place and might suggest checking your disposal to see if it is clogged or having other drainage issues as the water level in the disposal shouldn't ever get high enough for water to backwash down the to the dishwasher.
Also running the dishwasher into the disposal is beneficial so any large food waste goes through the disposal instead of directly down your drain so I wouldn't change that part out.
I would also double check your manual for your dishwasher to see if it has special instructions for the connection without the airgap.
Best Answer
Depending on what's behind that cover plate, you will probably be able to run some NB cable, if it' allowed, from there down the inside of the wall to a new junction box for the washer. If conduit is required, you'll have to cut away the wall below the counter to install your conduit. You cannot just change a 15 amp breaker to a 20 amp breaker without replacing the 14 AWG wire with 12AWG wire. If instructions call for a 20 amp circuit, then you'll have to run new wire. You'll have to check to see if GFCI protection is required. Since your basement is finished, you could use conduit or some of the decorative wire channel to run the wire on the surface of your finished wall. See picture below from Lowes.