usually they will exempt certain categories of employees related to "health, safety, security" and will limit the numbers furloughed (although those who do have to work don't get paid).
In other cases, if the "FTEs" (Full Time Equivalents) are being funded from sources other than "BA" (Budget Authority) money like "User (or other) Fees", "Excise Taxes", etc., and then those staff can continue to work (and get paid) until that funding runs out.
In general, any contractors are SOL (unless their prime chooses to eat the cost to pay them when the shutdown is done because they normally won't get any back pay and there are MANY MANY contractors that make up the federal workforce, and that would include DHS, the biggest department outside of DOD).
(image from here - https://www.brookings.edu/articles/is-government-too-big-reflections-on-the-size-and-composition-of-todays-federal-government/)
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