If you have an employee and taxes are greater than $2500, you have to file electronically?

Federally, there doesn't seem to be a requirement, at least as far as I can tell; in [Publication 926](https://www.irs.gov/publications/p926#en_US_2020_publink100086765 they state:

You’re not required to withhold federal income tax from wages you pay a household employee. You should withhold federal income tax only if your household employee asks you to withhold it and you agree. The employee must give you a completed Form W-4.

For employees in general, the requirement to submit taxes withheld is indeed at a minimum of $2500 per quarter, per Publication 15-T; that indicates that you must pay via EFT (EFTPS, specifically, in most cases); but household employees don't have the withholding requirement.

Massachusetts taxes would work the same way.
According to the Massachusetts Household Employer Tax guide:

State income tax withholding is not required for a household employee, but may be paid if the employee requests it and you agree.

and

Employers withholding $5,000 or more have to file returns and make payments electronically.

You do so by paying at Mass Tax Connect, which also has links as to how to file.

It seems like a rare case that a single household employer would need to withhold that much; you'd have to have several household employees to have that much required withholding. In the case that you can afford well over $100k in household employee salaries, you're not going to get much sympathy from me or anyone else if you also have to hire a tax accountant (and probably you should do that for other reasons in any event, at that income/wealth level).