Sounds like your secure hosting is misconfigured.
SNI has been available at DreamHost for quite some time now, and zillions of DreamHost sites are operating over HTTPS thanks to SNI.
Just for fun, I searched Twitter, found a few sites vying to be DreamHost site of the month, verified they’re on shared hosting, and ran them through the Qualsys SSL Labs checker. Most sites I tested got A grades and the checker reported “This site works only in browsers with SNI support” – exactly what you’d expect to see.
The problem you’re having is with how your site is configured, not with DreamHost’s server stack. Ask support to assist you by launching a live chat session or filing a ticket.