The phrase "higher minimums" means that the minimum requirement or standard has been increased or raised to a greater level than before. Full definition
Both houses also agree to increase money to direct care workers at not for profits who care for the disabled, to help meet new higher minimum wage requirements. (wxxinews.org)
This can happen if the primary user doesn't have enough money for higher minimum payments. (wisebread.com)
When you start looking around for a good savings vehicle, you often come across student savings accounts with high minimum balance requirements, and even higher fees. (myuniversitymoney.com)