Yes — Mbabane is generally safe for tourists who exercise normal urban caution. Eswatini has one of the lower violent-crime rates among Southern African capitals, and the city centre is walkable during the day without significant concern.
Petty theft (bag-snatching, pickpocketing) does occur in busy market areas and around the bus rank, so keep valuables in a hotel safe, carry only what you need, and be discreet with phones and cameras.
After dark, take registered taxis rather than walking; the city is small and taxis are inexpensive. Avoid poorly lit streets on the outskirts at night.
Road safety is a more notable risk: Eswatini's highways see fast-moving traffic, and driving standards vary. If renting a car, drive defensively and avoid night driving on rural roads.
The political situation is generally calm for visitors, though Eswatini has periodic civil unrest related to pro-democracy movements. Check your government's travel advisory before departure and avoid any gatherings or protests.
Medical facilities in Mbabane are limited compared to South African cities — ensure your travel insurance covers medical evacuation, and bring any prescription medication you need.