An ID card for aliens would not infringe on native born or naturalized persons' rights.
An alien ID card would have to be interactive and require check-in at frequent intervals - we've probably got that technology already.
Such a card would not solve the problem - but it would provide a basis for other actions such as,
An EO requiring that everyone here on a visa etc. obtain a card within 14 days - failure equals depotation when & if cought.
Authorizing local authority to arrest and detain illegals who do not comply - fed's could pay a carrying cost to the locals.
Advise our "friends" overseas (or the river) that all costs of the program will be prorated out of any foreign aid they might have expected - AND that repeated offenses will have a negative on aid they might want in the future.
Mandatory sentencing of illegals when cought at any crime - starting with seizure of assets and deportation of the offender and immediate family. (Capital crimes to be dealt with with respect for a "speedy trial" without recourse to appeal.)
Foreign officials found to be aiding or encouraging illegal immigration to be tried in abstencia, declared to be terrorists themselves, and their home governments fined as indicated above.
None of these suggestions involves "profiling" as such - they do recognize that the US has an obligation to (a) its CITIZENS and to those who come here for (b) legally, for (c) valid reasons, but not necessarily to the world's flotsam and jetsam.
Here's the problem with that approach - what if someone just refuses to use the card after they enter the country? Think about it. How are you going to tell the card-carrying aliens from citizens at a given checkpoint?