No, the dependent student would not get the $12K standard deduction, unless their earnings and/or scholarships came to $12K. If the student earnings/scholarships were $5000, then the dependent student would have a $5350 deduction. New tax law doesn’t change the deduction for dependents.
Obviously a student doesn’t really need a deduction that is more than they earn, if the only income is frrom earnings and scholarships. But remember this whole thing comes from tax rules meant to prevent parents from transferring assets to their kids to reduce the level of taxes on investment income. So if the kid also happens to have a trust fund that is yielding $10K a year in interest and dividends on top of the earnings & scholarships, then the kid is going to be taxed on the trust fund income. And for that the kiddie tax definitely kicks in and under the new law that means that the trust fund kid will be taxed at the rates set for estate income.