In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering what is the situation on this question today.

The monoid axiom for symmetric and orthogonal spectra of spaces is Proposition 12.5 of Mandell, May, Schwede, and Shipley's paper ``Model categories of diagram spectra''. 


I would look at Stefan Schwede's articles. I think the answer is yes, see page 7 of Schwede and Shipley: http://www.math.unibonn.de/people/schwede/AlgebrasModules.pdf Perhpas this more explicitly states what you want: http://arxiv.org/PS_cache/math/pdf/9803/9803002v1.pdf It even references the paper above. The dates in the bibliography may be a bit confusing. It becomes less confusing when you realize that Symmetric Spectra had been around for a year or so before the published version appeared. Certainly people in the know, such as hovey, schwede, and shipley, knew about Symmetric Spectra. If you tried really hard you could probably find seminar talks about such stuff. 

