There are ones that work down to DC, but they are usually called something different like a Dynamic Signal Analyzer and usually work in the digital domain.
That being said most spectrum analyzers will handle a little bit of DC on the input just fine. There is usually some sort of filter in the way that has DC blocking properties, but this being RF the components inside tend to be really tiny and delicate so they won't handle a whole lot of it. However there are some spectrum analyzers that WILL for sure get blown if you send DC into it. Some older high performance units have the mixers diodes sitting right on the input, since getting deep into microwave bands back then was hard. For this reason i have a DC block sitting on the input of my HP 8566B. But even a DC block has a certain voltage rating and so will blow up at some point, likely sending all the DC trough it. Even then the huge voltage step of touching a large DC voltage with a probe might produce a pulse large enough to damage the input.
If you do often need to probe DC voltage rails then you can still DIY yourself a probe for the task. Build the DC blocking and overload protection into your probe, test it to its limits on an oscilloscope so that you know it works, then plug that into your spectrum analyzer.
That being said the lower performance spectrum analyzers that only go into the low 100s of MHz are pretty resilient because much less RF optimization is required inside. They are also cheap on the used market and are easier to repair due to having less fancy special nude virgin manufactured RF black magic components inside.