Insight servers are open, but you need to find them. Simple curl requests will do.

Stratum servers (they power electrum) do the same, but you can find a list of them using IRC. Stratum offers SPV proofs for a transaction, so you can prove it was in the blockchain and they're not feeding you lies.
I wrote a PHP library for working with stratum servers. https://github.com/Bit-Wasp/stratum-php

It might be worth writing a library in JS to talk to stratum servers, I haven't seen one yet.

Edit: Avoid using the BC.I API. They've had extraordinary number of problems. Just use something that talks to a full node.