Article 3 of the Constitution provides:
In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.
So far as I can tell, the Eleventh Amendment does not modify this in a way that makes a difference in this particular case.
Washington is a state, and it is party to Washington v. Trump. Why is the lawsuit currently before a district court instead of SCOTUS? Does Washington have the power to decide the venue? If not, could the Supreme Court invalidate any rulings of the lower courts as lacking jurisdiction?