The Browning designed M1903 has both a manual safety and a grip safety (later ones also have a magazine safety), so it's a pretty safe weapon. However, there is nothing that blocks the firing pin, so while it probably will not go off if mishandled, in the event of a parts failure, it could unintentionally fire.
I think the chances of that are vanishingly rare, but something to consider when it's in your pocket pointed at the family jewels...
Additional issues are whether it is reliable with modern JHP ammo: You should test fire your desired carry load extensively. Also, an older gun could be damaged with extensive use of more powerful "+P" type ammo.
I have a M1910 Browning that I used to carry quite a bit, and never had any problems.