Don't provide any password to a border agent, or really anyone who doesn't need it.
My company is currently in the process of designing a special TPM style product that makes it very near impossible to enter a devices without being the one intended for reception. Well solutions like this do exist, ours is going to be fairly open, cheap and allow it to interface to almost device to which someone can write a low level kernel based driver. With our device, it makes it impossible to access the contents of anything on the device under encryption due to how the data is stored and decrypted. Without access to the exact key which is paired to the device under encryption, you may as well wipe the device because except in exceptional cases, where multiple keys are warranted, there is no other way in the device under encryption.
I'm bringing this up for this exact kind of situation, well traveling you can keep your data fully encrypted, have one of our keys at home, with the data it encrypted being unavailable physically until you arrive home, and you could carry a second key which can decrypt any data marked for use between the two keys or just the data encrypted well traveling, with the only way to view the date, to be in possession of a key physically, think very small USB thumb drive.
If the border needs access, they can get access themselves. You're not stopping them by giving your phone, and you're not stopping them by refusing to give up a password or encryption key, you're simply protecting your right against possible self incrimination, and if the border patrol is actually qualified in the first place to do a job that would be require decryption information on a phone, they should be able to do it regardless of what you put on it. I know that's a ridiculous statement, but it works. You shouldn't have to provide access to your personal data, to anyone. If anyone wants access, they can get access themselves without you.
I even once gave the border an entire database encrypted with our key solution, told them how it was encrypted and that the key for decryption was already sitting at an office in the US, so even if I wanted to get the data, I couldn't, they had no choice but to let me travel. You're not blacking anything by refused to decrypt data or let them into the system. In our case, we're going to the Nth degree and making it a physical problem, where it doesn't matter if you know the password, because it's point to point tied down.
I support anyone who refused to give up access, it's the right thing to do, the access isn't theirs and if it is, they can enter it themselves.