As an employer, what taxes are supposed to be withheld for J1 Visa employees (just here for the summer) - with or without a social security number?

Every one on J1 visa is exempt from residency for two year that is the income is not subject to FICA (Social Security and Medicare) taxes. Income is subject to income taxes.
J1 must file nonresident tax return. Will get personal exemption and itemized deductions (no standard deduction).
Your U.S. Tax Return: U.S. Tax Filing Requirements for Non-Residents

The taxes that should be withheld are federal and state income tax (assuming that you are working in a state with a state income tax).