Chose based on looks or stats. Mithra will have lower STR and VIT, but high DEX and AGI so we won't hit as hard as everyone else, but we will hit more accurately and evade more. Galka have high VIT, STR and HP so they can take more hits and do a little more damage, but lack in the evasion. Elvaan have highest STR and good VIT, but are lowest in AGI, DEX, and INT so they aren't as accurate or evasive as other races. Taru are low HP, VIT, and STR so they can't take or give hits very well, but have high AGI and I think DEX so they can evade well. Hume are basically just average and have none of the highest nor the lowest stats which make it a good choice for someone who wants to do everything.
NIN relies on a large amount of stats, a lot more than most other jobs. NIN is considered a tank which uses Utsusemi to create shadows to avoid taking damage, but need high evasion so that all shadows are not taken down before recast. So it's important to have a large amount of evasion/AGI gear to evade. VIT and HP help when your shadows are down and you are taking damage, but most do not rely on this type of gear. STR is often used as a source to keep hate as a tank. If you can deal as much damage as other melee, you should be able to hold hate relatively well. In later levels you will get +enmity gear which will help to keep hate, but at early levels it's not a large option and most NIN rely on their own damage to be able to tank effectively. DEX is for all melee and helps with accuracy and critical hits. Although the boost isn't vastly significant, but sometimes it's better than nothing. INT helps with the Ninjutsu elemental spells and maybe the debuffs as well (I've never been sure or found out if MND and/or INT effects these. Anyone know for sure?).
Although the stats help in a lot of cases, it's not always a huge issue. There are a lot of pieces of armor and food that will boost your stats that will allow you to do your job effectively regardless of race. The only real exceptions are the ones Mishaela stated which can be troublesome, but people have learned to deal with it regardless.