... You're renting an apartment sight-unseen in another country?
Is this university-affiliated housing, or just a place you found online?
It should go without saying, but just in case: if this is not university-owned housing, don't send them any money until you've seen the apartment & confirmed its condition/legitimacy.
Do you have a bank account w/a debit card you can use internationally?
Does it have a daily limit or per-transaction limit? Does your bank know you'll be traveling internationally and can they temporarily raise those limits? (Otherwise they might very well lock your card entirely for an unusual charge).
If all you have is cash, then it would be safer to buy an International Money Order as mentioned in comment above. It should be made out to the recipient, but still keep it secure on your person while traveling. If you're carrying less than USD$10K, you do not have to declare it at customs.
If you have a debit/credit card, you can buy money orders directly w/that upon arrival at larger grocery/retail stores (WalMart, Target, Safeway, etc), or a U.S. Post Office. Or, withdraw the cash at such a store & buy them right away so you don't carry a wad of cash (Note: Post Offices usually do not have ATMs for cash withdrawals). This may be preferable to having to deal with it ahead of time & keeping track of it during international travel.
Money orders often have a small fee and a dollar limit as well, so you may need to buy 2-3 separate smaller money orders to reach the total you need. If that total exceeds any daily transaction limit on your account, you might need to do this over multiple days. Even if paying cash, you might wind up purchasing multiple smaller-value money orders depending on the total value you need.
Note that a money order should be made out to a specific recipient to avoid being cashed by other people, but cannot be replaced, so if you lose it or it gets stolen you are stuck. Given that, I'd suggest it's probably best to wait until you're in Kansas, have seen/verified that you want to rent the apartment, and then get money orders just before going to sign your lease.
Does your university have an affiliated credit union or bank? Presumably you're already planning on doing so, but open a local checking account upon arrival so you can (ideally automatically) pay future rent and utilities from that. You will likely need your passport, student visa, maybe some minimum deposit amount, and possibly a statement of employment/income/enrollment to do so.
Hopefully Kansas will be good to you, and you'll have safe travels and an enjoyable time in grad school. Welcome.