I want the subscription to start today but allow the customer to choose which day the payment will be made in subsequent months. Stripe allows this with the 'billing_cycle_anchor', but I am not sure how to do the same thing with Authorize.net.
To do this is a two step process. First you will process their first payment immediately (which you should be doing anyway so you can capture that initial payment and confirm the credit card is valid as well). Then you can create the subscription using any date you choose.
You should keep in mind that if someone were to establish a subscription today (the 19th of the month) and then choose to have their subscription start on the first of the month, you will need to prorate their first payment so they are only paying for the remaining 12 days of the month.
So, if your monthly subscription is $10, you would only charge them 39% of that $10 which is $3.90. Then their monthly subscription would start on the first of the next month at the full $10 subscription price.