The safety risk isn't necessarily an insurmountable obstacle; I could imagine the U.S. government being more likely to approve sending a nuclear reactor into space if it was NASA that was asking as opposed to a private company, but either way it's more of a political problem than a technical one.
Nuclear rocket engines are tricky. Nuclear reactors don't actually work very well in space because there isn't any convenient way to get rid of the excess heat. A vacuum is a very good insulator, so usually your only option is just to radiate it away as infrared light.
With a rocket there's another option which is to transfer all the heat to the reaction mass you're expelling out the back of the ship. That sounds like a hard engineering problem though.
Using a reactor on the Mars surface is a lot more straightforward because you can use the local air and ground to transfer heat. And since Mars is so cold, you might even get better steam generator efficiency there than on Earth, where the ambient temperatures are higher.
One of the hopes with fusion is that if it pans out it might be reasonable to send a fusion reactor to Mars since you wouldn't need to send radioactive fuel rods. In fact, maybe the first practical fusion reactors will be used on Mars before they're used in more than a demonstrative capacity on Earth because they fit a very specific need, there are barriers to using the alternatives, and cost per kw/h isn't the most important constraint.
The US did build a nuclear powered ramjet engine and test it. This isn't an insurmountable hurdle, it's an achieved one. The design they used of course has lots of good reasons we should not use it on Earth (namely, the reactor would activate the air, meaning it spews a plume of oxygen and nitrogen isotopes) but it does work.
You can solve the radionucleotide problem by running it off hydrogen instead, which doesn't activate in any meaningful quantity.
Basically, not only do we have the ability, we had it in 1961.
Interesting. That requires an atmosphere though; it might work for the first stage of a launch from Earth, and maybe even for Mars. Once you're out of the atmosphere you'd need some other method of propulsion.
I think nuclear space propulsion would be used as a reactor generating energy for ionic drive. That would work as a Mars-Earth shuttle while never going into atmosphere. The landing to and launching from the plane surface would be done by regular chemical rockets.
Specifically it requires propellant, which on Earth is the air intake. For space travel you have to still take a propellant, but it can be essentially any type of gaseous reaction mass - for example, water.
Nuclear rocket engines are tricky. Nuclear reactors don't actually work very well in space because there isn't any convenient way to get rid of the excess heat. A vacuum is a very good insulator, so usually your only option is just to radiate it away as infrared light.
With a rocket there's another option which is to transfer all the heat to the reaction mass you're expelling out the back of the ship. That sounds like a hard engineering problem though.
Using a reactor on the Mars surface is a lot more straightforward because you can use the local air and ground to transfer heat. And since Mars is so cold, you might even get better steam generator efficiency there than on Earth, where the ambient temperatures are higher.
One of the hopes with fusion is that if it pans out it might be reasonable to send a fusion reactor to Mars since you wouldn't need to send radioactive fuel rods. In fact, maybe the first practical fusion reactors will be used on Mars before they're used in more than a demonstrative capacity on Earth because they fit a very specific need, there are barriers to using the alternatives, and cost per kw/h isn't the most important constraint.