I agree with the previous comments - we are a conservative profession and that is how we perceive ourselves and are perceived by the general public. Tattoos do not fit into that mould (nor for that matter do dyed hair, earrings on blokes, nose rings etc etc). It is not a moral issue but it is one of acceptability. The average passenger only wants to see a pilot with a dark uniform, short hair and no earrings or tattoos. You may not like it, but that is the reality of the situation. We are in the business of providing reassurance and not in the business of challenging society's attitude to looks and impressions. In terms of looks we are expected to be conformists for the sake of our passengers.
Added to that, most pilots are from a middle class background. Once again, whether you like it or not, it is true to say that people from that background by and large do not approve of tattoos, noserings, male hair dye and so forth. Just remember that it is these people who will be deciding your future in a very competitive industry for the foreseeable future. This is not meant as some sort of political view but merely a statement of the harsh realities. You would be very unwise indeed to ignore these realities.
Finally by way of anecdotal evidence, we have a great pilot in our company who is an ex-Royal Marine. I noticed recently that he had a fine set of tattoos on his upper arms and I made some joke about them. He was a bit sheepish that I had even noticed them and pointed out that he now feels somewhat embarrassed at having had these tattoos done and wishes he had been more sensible in his youth (his views not mine). Perhaps a warning for those with a short term view. Once you have them they are there to stay!