Installing a dishwasher where there is no preexisting one
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.