There should never be amnesty even after the borders are secured. The Constitution forbids all Ex Post Facto laws (at both the federal and state levels). This means that it is illegal to pass a law which would change the punishment for a crime that has already been committed. (It works both ways--you can’t legislatively increase or reduce the punishment.) Immigration law can be changed so that future violators are not punished as harshly, but whatever punishment was prescribed at the time that the violation occurred, must be carried out or else the government is derelict in its duty. The Constitution does give the Congress the authority to restrict immigration.
This does not necessarily mean that there cannot be “a path to citizenship” for illegal immigrants as long as it is allowed by the law that was in place at the time that the violation occurred.
The only other *possible* Constitutional remedy is for the Executive to grant pardons to the violators, but this must be done one at a time. (But I am not sure that the pardoning power that the Framers had in mind really goes as far as we have taken it today.)
I am a believer in freedom for all. I do not believe that crossing a border is evil except in extenuating circumstances (e.g. for security reasons we must keep out violent criminals). I do not believe in punishing people for something that is not evil. (The Bible says that you will be judged by the same measure that you judge other people.) Even though the Constitution gives the Congress the authority to restrict immigration (for virtually any reason), that doesn’t mean that they have to exercise this power for no good reason. There wouldn’t be an illegal immigration problem if immigration hadn’t been made illegal in the first place! If an honest person wants to come to America and give an honest day’s work in exchange for an honest day’s pay, then why not let them do it?
And yes, of course we do need to secure the borders because the Congress is required by the Constitution “to protect the States from invasion”.
The other problem with this issue is "birthplace location citizenship". This problem is caused by the so called 14th Amendment and by the outrageous interpretation of its phrase "under the jurisdiction thereof". Requirements for citizenship should be much higher than the requirements to immigrate into the U.S.