It's obvious that it's impossible to stay current with all of the literature. The deluge of papers was a common complaint even back in the 1960s. Abstracting services like ISI started up to help handle that data flow. I've decided that it's not worth stressing out over it.
There are two primary journals for my field, cheminformatics. These are the "Journal of Chemical Information and Modeling" and "The Journal of Cheminformatics." There are several other journals which will publish on cheminformatics, such as "Algorithms for Molecular Biology."
I have subscribed to the monthly email announcement and/or RSS feeds for those journals.
I also learn about what's important by watching the blogs of others in my field. I going to conferences to see what people are using. For example, one conference I went to 1.5 years ago had several papers on using "Matched Molecular Pairs" (published in JCIM). I talk to people, and see what they are interested in ("Activity cliffs" being another recent hotness).
I'm very demand driven by what I read in depth. I scan the abstract, and only read further if it's something that interests me deeply. This could be something that I've worked in before, or something that I've thought about but haven't seen much in-depth work in. I'll mostly go into a paper when I need to do something related to that topic, which is also how I handle papers outside the field.
The field of course draws from many other fields. For those I use the approach that I do for buying computers: mostly ignore things but every once in a while read a paper and make sure I understand the key concepts. When I need to get into that field, dig up the literature and work on a task.
For example, one of the things I'm interested in involves postings lists. This is classic information retrieval, which I knew about from reading "Managing Gigabytes" and "Mining of Massive Dataasets." That was enough to be able to understand the literature when I did a search last summer.