Greetings,
not to worry, climbing upstairs will not harm you.
Next Steps
if you are used to doing exercises before pregnancy, then do exercise to strengthen pelvic floor. if you are new to exercise then hire a physio to exercise your body. uterus is strong to hold any load on it, so just make sure structures around it are strong. climbing stairs with weights should be avoided.
Health Tips
maintain proper diet plus being in calm state helps you build strong body as well. if any other questions you may have, do let me know.