In Figure 1 from this article about the Open DMAP information extraction system the authors suggest that, as of 2007, about 40% of the articles indexed in PubMed contained some information about genes that could be extracted from their abstract. That seems like a lot.. If anyone has any data to corroborate or to disprove that statement, please let me know.
--Correction. The article said 40% of recent articles had a gene mention. See comments from Joachim below that suggest a better estimate overall of 19% for articles with abstracts.--