I published a paper late 2016. Recently I noticed that the published manuscript has a typo error in one of the authors affiliation. But it got indexed correctly on all databases under that author. Will it be good to contact the journal now to rectify the mistake or should I not care about that much as anyway it got indexed correctly? If I contact the journal almost after an year, will it affect the reputation of the authors? Also one of the student is going to use this paper for his PhD thesis. Considering this fact should, what is the best strategy to handle this ? Any recommendations?
If the error is significant enough to make it difficult to identify the institution involved, then your best bet is to send a correction to the sites that handle the indexing. Going through the journal will require a corrigendum and won't really solve the issue.
I suggest doing nothing. Once a paper has been published, making further changes is a difficult thing. With electronic versions it's easy to correct the typos, but since the electronic papers have already been disseminated, lots of copies of the paper with the typo included are already worldwide. The publisher cannot recall those copies, fix the typo, and then send them out again. With paper versions there're obviously even more problems since implementing the correction is hard.
If you must make a correction, it depends on how severe it is. If it's very severe, enough so that the veracity of the results are threatened, I would contact the publisher and ask to issue a corrigendum or erratum (see this link for more details about either). But in this case it's just a typo in the affiliations. This typo makes no difference to the research conducted or the results. Nobody is likely to care, especially not the researchers who will be using the work. Many might not even notice.
tl; dr: relax! There are far better things to spend your time on.