Hi Simon,
I can't comment on AuthorizeNet, but we have been using Stripe with this plugin: https://moodle.org/plugins/paygw_stripe for a number of years without any issues and can definitely recommend it. We found Stripe very easy to set up (account setup and verification took a mere 2 days!) and to deal with on the very rare occasions we needed to contact them. Their transaction fees are also very reasonable (at least of us in Australia and New Zealand). We've never had any issues with course payments or with the payments deposited into our bank account.
I can't comment on AuthorizeNet, but we have been using Stripe with this plugin: https://moodle.org/plugins/paygw_stripe for a number of years without any issues and can definitely recommend it. We found Stripe very easy to set up (account setup and verification took a mere 2 days!) and to deal with on the very rare occasions we needed to contact them. Their transaction fees are also very reasonable (at least of us in Australia and New Zealand). We've never had any issues with course payments or with the payments deposited into our bank account.