this industry needs to improve card security...some simple measures would greatly allay these consumer fears:
1) remove the PAN from the card...it is a private-key...a password...nobody recommends putting your PIN or password anywhere visible...why are we still doing this for cards? Rev-Money cards didnt have PANs and they worked fine. Remove the PAN.
2) EMV everywhere. Its not the be-all/end-all but its better than magstripe. Just do it...pick up the pace.
3) Toggle on/off: enable users to toggle their cards on/off with a simple swipe of an app...yes, it'll trigger a ton of unintended declines for a while...but, we'll get used to it...and, its a gateway/step process to just using the phone at POS...even ApplePay requires the user to instruct the phone to be ready to pay
4) $0 liability everywhere and everyone knowing it--make cards as disposable as prepaid SIM cards
5) instant replacements at banks...maybe even retail (link new card to old account via cloud service)

Finally, offer real value around using a phone to pay at POS...and real merchant coverage...so, if you lose your card, you can still spend with your phone at most places you would have used the card