Most people who are afraid of getting tattooed in Bangkok or any other Thai city is due to bad press and generally non-sense fear. But well, you have to know that most tattoo parlors here use the same hygiene standards as studios in NYC, London, Tokyo, etc. You get the picture: there's nothing to worry about if the parlor is good and reputable. With the internet you can quickly tell if a studio is good or not, because you can look for reviews online...
However, when I arrived here I didn't have a single tattoo lol, but I got inked for the first time in All Day Tattoo Studio, and the experience was pretty good because as I was pretty noob, they helped me through the process from choosing my tattoo to advice on how to handle the post-healing process. So yeah I can vouch for them, and their prices are pretty good so they are a good option IMHO.
I also know that there are other good studios around and that the quality of tattoo artists in BKK is beyond average, as they are pretty creative, talented and experience. So just make sure the parlor is reputable and that's it! :)