When I first started playing, I didn't really pay much attention to what my influences were playing. My first bass was a Jazz knock off, because it's what was available in the local pawn shop. When I finally got about to buying my own, "real" bass, all I could think about was getting a 51 Precision, because I saw a picture of Dusty Hill playing one (he wasn't even my biggest influence at the time, I just liked the look of the bass). I headed down to the Guitar Center in Seattle, found the 51, played it, and decided I found it uncomfortable. I knew some of my influences (JPJ, Noel Redding, Geddy, Jacob Sproul) played Jazzes, so I tried out a Jazz, and I found the neck too small for me. The last bass I tried out, on suggestion from my father, because he liked grain of it, was a 62 AVRI P-bass. That's the bass I walked out with that day, and it's the bass I've played nearly every day since.
The next bass I bought was a Zebrawood T-Bird, because I loved the raunchy tone and the wood of it.
Now, my amps, THOSE I've bought because of my influences: Sunn 2000s, thanks to Noel Redding, Felix Pappalardi, and John Entwistle.
I can't say I haven't gassed for basses due to influences though: I really want a 64 EB-3, because of Jack Bruce and Andy Fraser, and a bound/block Jazz with either a Darkstar or Mudbucker due to Berry Oakley & Mel Schacher, or a Telecaster bass thanks to Ron Wood.
My next bass though will be an amalgam of those influences though, but I want to make it distinctly mine, so influences do shape my purchases to some extent.