As an occupation, the common terms are:
- fence builder
- fence erector
- fence installer
They are also mentioned under construction trades in "1990 Census of Population and Housing: Classified Index of Industries and Occupations" by U.S Department of Commerce:

Other related occupations that are mentioned:
- concrete fence builder
- wire-fence builder
- wire-fence erector
- wood-fence installer
- rustic-fence builder
For example, fence erector and fence builder are both mentioned in a job opening in Canada Job Bank:

OED lists fencer also for this sense and gives the earliest reference from 1881:
Chiefly Austral. and N.Z. One employed in putting up fences.
1881 R. C. Praed Policy & Passion I. 241 ‘Where is father?’..‘In the office settling with the fencers.’