It’s been one of the most discussed topics in SEO world for whom use WordPress for blogging or other businesses.
YES it affects the traffic and page rank. There are few causes and solution
1. Most themes are not SEO friendly , it seems most developers doesn’t know about the HTML title tag is followed by google search spider. Like you shouldn’t use more than one h1 tag. You must keep your Focus keywords in
h2 tags. also you should use h3 tag for subheadings.
2. It can affects your branding. But i suggest to change themes as version for example 1.0 , 2.0 so on in an interval may be a year or six months. I think it will excite your readers/customers for new feel.
3. To change themes you will definitely loose your Google Analytics, Webmaster tools and other tracking code for sure. So if you change themes weekly or monthly, i believe it wouldn’t be a good idea.
4. You will loose custom settings for your site that you need at most.
5. Some themes are heavily built like JS CSS are not optimized or zipped to load site quickly, so i suggest before using any themes permanently, active the themes and take a test on GTMATRIX to see how your site loading. If you see it pleasantly loading, then keep it up.
Here is the bottom line, i don’t like to change themes regularly, but after a long interval like an year or so. I know it will stimulate the visitors experience as they can find it user friendly. that’s the main fact. SEO and user friendly those should be followed by when you think of changing your themes.