(thumbs u
Oddly enough, both started coming into their own as major comic influences around 1981-1983.
Miller's defining moment of being seen as a major writing force is Daredevil 168 (January, 1981). He'd already received his warmup co-writing on the title, but now it was his show after convincing Shooter he could turn the character around from a dog to a major force.
Alan Moore had been writing and drawing sideline characters before getting involved in Doctor Who, Judge Dredd, and Captain Britain stories in the early 80's. Then, stories with Marvelman and V For Vendetta through Warrior Magazine.
So not sure if they influenced each other, but they really started taking off in that 1981-1983 period in their own markets, and would later grow into much more recognized creative powerhouses from there on out. Not sure if they were aware of each other at that point, but I doubt they influenced each other.
By the way, here is a very interesting interview with Alan Moore about his influences over the years. He mentions Flash, Action Comics, Detective Comics as major influences, though his concern was it was fantasy land where they lived. I wonder if this is why he wanted to create more realistic stories to bring characters into the world we all live in versus happy-peppy environments all the time.
http://www.comicbookresources.com/?page=article&id=511