Wow...I've tried lots of options. I used to mail DVDs. But TiredIron is right--a lot fewer computers with DVD drives. And despite a padded mailer, I've had some arrive broken or chipped.
Lately I tried Box. That's fine as long as you have high speed upload options. Here's the link to Box: Personal Product Overview | Box
When I was doing recent research on this, here's a decent article: Reviews of the Top 10 Photo Sharing Websites 2014 Now it's specific to photo-sharing (which isn't the same as file downloads) but there's some overlap.
I use Zenfolio for my proofing galleries. I don't allow direct downloading on mine, I have them choose their photos by adding them to a collection, then they can simply click "Send collection to photographer". Then I edit the selection, and email them their files which they can then do with as they wish. It's very flexible and you can set it up to allow downloading. You can even restrict the maximum size people can download and switch it to commerce if you like. It is quite pricey though, at about £100 per year. It is a very good system, though.
I have customised mine to match my main website and my clients have said they find it user friendly and fun to use.
I avoid putting my clients' photos on physical items like discs (my brand new Macbook does not have a disc drive) or USB drives which can easily be lost or damaged in the post.
I generally drop them off via thumb drive in person. Well, for commercial folks. I like the extra interaction, and get to collect payment then too. It works for all parties.