Stockport County chairman Lord Peter Snape says some of the club's directors have been the subject of death threats.
Stockport are 18th in the Blue Square Bet Premier table and only three points above the relegation zone.
"When we find it unsafe to bring our partners, children or grandchildren to home games, we are entitled to ask 'is it all worth it?'
"When some internet sites carry death threats against particular individuals, it certainly isn't," Lord Snape said.
However, he insisted that the club "will not hesitate to prosecute or ban" anyone using threatening behaviour.
The Hatters were playing in League One in 2009-10 but their financial troubles resulted in the club sliding into non-league football and they have only won once in their last eight matches.
after reaching an agreement with Cheshire Sport in November, but the club's budget remains tight and Lord Snape has asked for patience from supporters.
He added: "Whilst the board and club's employees fully understand the anger and frustration of many of our supporters and their disappointment with recent results, we are not prepared to accept the abuse, foul language and threatening behaviour aimed at the directors' box following the last home game
"Let me make it quite clear. There is no consortium waiting in the wings to take over from the existing board. Indeed, if the current board of directors walks away, there is no longer a football club called Stockport County.
"The board is grateful for the support and patience of the vast majority of our fans whose behaviour is exemplary and who, like us, want only the best for our club and its future."