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There are lots of journals that only publish online. However they are indexed by scopus and ISI so they are unlikely to be predatory journals. Are they safe? What are the chances that some hacker erases the journal’s database and all the records of publications are lost?
Surely it’s better to have printed copies of volumes and stuff. At least that way there is always proof that one’s article has been published.
Any reputable online-only journal will have a plan for data retention that goes beyond what's visible to the user. There will be backups somewhere.
Print journals may have the advantage of having a physical repository copy existing, but even those are not necessarily of "archival quality" (that is, printed on acid-free paper and using techniques that promote their long-term preservation).
It seems the ArXiv stores its data on Amazon S3 cloud servers. I would assume Amazon has a data safety and backup policy that is as professional as it gets.
Presumably also most journals don't store their data on their own servers, but use one of the large file hosting services.
Being online-only is orthogonal to the issue of journal quality. Any new, quality journal will almost always be online-only these days, for economic reasons. Any new or existing predatory journal will generally be online-only, for economic reasons. Probably the best working guide to a journal's quality is "Do I read and cite papers from this journal?"
On the issue of archival security of publications, you should look for explicit statements on the journal web page about that. An open-access journal that takes this issue seriously will have plans that involve archiving content on at least one third-party, independent, and ideally distributed, service. Some other answers here have mentioned technical issues, such as using well-regarded storage services such as Amazon S3. But the quality of such services is not relevant if the company using them goes out of business and no longer pays to maintain the service. Archival security should persist even if the publishing entity itself does not.
For example, many open access journals in the medical field will archive their content with PubMed Central, a service of the US National Library of Medicine of the National Institutes of Health. Until recently, that might have seemed like a gold-standard of stability and security, so additional facilities that are international, collaborative, non-profit, and ideally distributed might provide additional peace of mind. For example, look at the CLOCKSS and LOCKSS initiatives.
An example of an online-only, reputable journal that pays attention to independent archiving is PeerJ. Their page here gives an example of the attention paid to these sorts of policies, which you might compare with when seeking a journal in your field that takes these issues seriously.
Such information is not always prominent on a journal website however. In a comment to another answer, user @Emilie points to a very useful central resource: "the Keeper's registry, that tells you WHERE the archive (or backup) of a journal are: thekeepers.org".
