The short answer is that the 2018-2019 US government shutdown shouldn't affect student loans for the vast majority of students, but some may have some issues.
Now, for the longer answer:
First, note that the Department of Education has contracted out loan services. In effect, the loan payouts (and repayments) are administered by non-government entities. These will still be disbursing grants and federal student loans. Second, I believe student aid programs are forward-funded to cover an academic year, rather than the calendar year, so the Spring semester should've been funded already. Third, the Department of Education is actually not affected by the 2018-2019 shutdown in the first place, as Congress has already approved the agency's spending bill.
That said, it turns out that the partial shutdown of the Internal Revenue Service (IRS) can cause issues for some. In particular, obtaining income documentation directly from the IRS does not appear to be possible at this time. This can affect both people repaying already incurred student loans, and students who applied for federal student aid during the shutdown - particularly lower income students, who are more likely to be flagged for income verification. So, to connect back to your question, the shutdown can affect students who didn't rely on federal loans or student aid during the Fall semester, but applied for the Spring semester, and who didn't have all necessary documentation available.
I don't know how many graduate students that would apply to, but it certainly applies to many undergraduate students. There have in fact been news stories about it. Now, I expect most universities will be flexible in how they handle these situations, perhaps accepting some forms of documentation they otherwise wouldn't have, or extending tuition deadlines. Students who find themselves in this mess should probably speak to their financial aid office.