Heard that Zika virus has spread to Bangkok. Is it safe to travel there right now?

For me personally, as long as I take precautions, I wouldn't be too worried. Bangkok is generally clean (assuming you are going to Central/Business district). Wear covered clothes when you're out and stay in a trusted clean hotel away from main drains. Put on repellants if you're going outdoors or bring "Clip on", a repellent that you just have to put it somewhere you and it provides you with day long protection. I don't believe in changing my plans just because of something that could be prevented. Just like the MERS outbreak, take precautions and go about your daily life.