Mr Shanmuganathan, who also has additional charge of Arunachal Pradesh, has denied the allegations. His resignation has been accepted.
The letter signed by 98 workers of the Governor's House in Shillong - widely circulated on social media - lists a series of misdemeanours including sexual misconduct and "torture and humiliation" of officers and staff.
"From the time he has taken over, employees are going through severe humiliation, mental stress and torture," the employees write.
The 67-year-old was accused of hiring or posting only women in his staff and turning the Raj Bhavan into a place where "young ladies come and go at will with direct orders from the governor."
The letter said: "His activities hurt the decorum and prestige of the Raj Bhavan as well as the sentiments of the Raj Bhavan employees who are very much attached to the Raj Bhavan, it becomes our concern. From the time he has taken over, the employees are going through severe humiliation, mental stress and torture."
Mr Shanmughanathan was appointed Governor of Meghalaya in 2015 by the central government. He is expected to fly to Delhi today.
"Till now there are no statements from any women. Facts should be looked into before commenting," Nalin Kohli, spokesperson of the ruling BJP, said today.
Protests by civil society groups erupted on the streets of Shillong yesterday as Republic Day was being celebrated. "The Governor should step down. We want impartial probe into sexual harassment in the governor's house," said Tarun Bharatiya, one of the protesters.
Chief Minister Mukul Sangma had said he was waiting for a decision from PM Modi and the Home Ministry.