On Dreadnought acoustic guitars, would deep bass response strings work best or normal size strings ?

I've never heard of “deep bass response strings” - I don't see how that would work. The tone comes more from the guitar than the strings. 

A dreadnought usually has a fairly bassy sound anyway - most people wouldn't want more of it.

I'd say use ordinary strings on any acoustic guitar - whatever gauge you like, coated if you want them and then either phosphor bronze which has a warmer sound or brass which has a brighter sound.

It depends on your playing style, I expect. 

"Deep bass response" sounds like you mean acoustic medium gauge strings.  These usually have a string gauge of about 0.056 on the low E string.  Bluegrass players tend to use these because they usually play unplugged or miked as a group.  The extra string mass generates and holds more energy and that means more volume to "cut through" the band on a lead or fill that needs to stand out.

Heavier strings require higher tension to tune up to standard pitch, so you may need to get a professional setup done to make it's not too much stress.  You could also tune down to Eb or D to reduce the tension and then capo back up when needed to play in standard pitch in open position.

You might find those strings harder on your fingertips, too.