It depends on the funds you invest in.
Load funds (those that charge a fee to invest) will charge that fee on each monthly investment. Some load funds may have special share classes that have reduced fees. You will have to research the fund (via its prospectus or a trusted financial advisor) to determine if you qualify to use such a share class.
Some funds (and some share classes) charge smaller fees when you invest, but may include a contingent fee that will be paid when you sell. They're contingent upon the time period you own them. After some period of time, the fee is usually removed. Again, read the prospectus.
No-load funds do not charge a fee to invest, and so they would be a good place to look, especially for index funds.
Most brokerages do not charge an additional fee to invest in mutual funds. They are paid a commission by the fund, through the sales fee and annual expense charges of the funds.
If you're investing in index ETFs or closed-end funds (which trade on exchanges), there is likely a commission to be paid for each monthly investment. Your specific brokerage may have a special arrangement for recurring monthly investing. You'll need to research that with your brokerage.