Recently, a newly published paper in a journal cited one of our papers and managed to misspell my name in the reference list. How should I deal with this? Simply ignore this? Or should I contact the authors or publisher to mention this?
In addition to sending an e-mail to the journal, as suggested already, you should check the most common academic databases such as Scopus and Web of Knowledge/Science and submit corrections to them so that they pick up the citation correctly. They have a form to submit this kind of corrections automatically; it will then take a few weeks to apply them.
People often (way too often, actually) rely on these databases to measure the performance of academics, so it is important that they don't miss any of your citations.
There are a number of services that assign unique identifiers to authors and then associate publications with a particular author. This is particularly helpful for identifying publications of authors that have common names (“John A. Smith.”) For example, Thomson Reuters (who publish the “Web of Science” database) have “ResearcherID”. Another important service is ORCID. You should register yourself with these services and establish your connection to your papers. You should also decide now on one version of your name (e.g. whether to use a middle initial or full middle name) and try to stick with it through out your career.
It may or may not be possible to get the journal to correct the spelling of your name in the online version of the paper. It certainly will not be possible to get this fixed in printed volumes that have already been distributed. That is why it is particularly important to check this information in reviewing page proofs before the paper is published.