This is where the religious wars about cabling specification start to escalate. It isn't your access point or your router that are affected by the lights, it is the wire. Wire acts like an antenna to pick up signals from other sources.
Basic Category 5 network cabling is very sensitive to the electro-magnetic interference (EMI) from fluorescent ballasts. Your cabling, whether to a wall jack or router or wireless access point, needs to be kept away (18" or more if possible) from those fluorescent fixtures, not run in parallel with power lines (Romex, etc), and properly terminated on quality jacks with minimum loss of twists and no exposed bare wire.
Standard Cat6 wire is twisted tighter and a little less influenced by EMI, but not immune by any stretch of the imagination.
The newer shielded twisted pair wire (Cat6a) is far less prone to EMI and more tolerant of closer (not close) proximity to power lines and ballasts.