Do not use Coinbase! They have extremely poor service and will not support you if you ever encounter issues logging in. Instead, they will write a generic response to your ticket and immediately close it. If any major bank tried to pull this they would get buried in lawsuits.
Again, avoid this service at all costs. Forget their greedy 5% fees on deposits/withdrawals - it doesn't even matter if you can't get your BTC or USD back period. That's a 100% fee.
You may share your problem with us. If there is any possibilities to educate you in login issues means we are happy to help you. But not sure why this big exchange does not reply with the customer service.
When there is investment fund on any exchanges are wallet you must keep it safe with 2FA and SMS auth if possible. Apart from that you may choose the wallet with the private key secured to keep it safe.