Bicycle designed for one-arm usage?
There are devices that allow a bicycle to be modified to make it easier and safer to ride with one arm. Riding safely would come with practice and confidence.  Trikes or recumbents mentioned in the comments offer a safe alternative also.

The key modification needed for a bike is a steering damper. The damper adds resistance to steering by way of a spring and helps slow down movement. It relieves the twitchiness of one armed steering. Steering dampers vary from a rudimentary spring based system such as the Hebie 695
a more sophisticated elastomer spring like the Hebie 696 

or the hydraulic Hopey system. 

Other modifications include mounting both brake levers on one side to allow single hand usage and using 1x gearing with a single shifter. (or in the case below a 3x9 in an over/under shifter configuration)



There are a few online resources such as NotBroken which show both modified bikes and riders who use a single arm such as Tom Bannister.