The prevailing practice is to say the bracha but without the name of God. So skip the parentheses of
baruch (ata Hashem elokeinu melech haOlam) she-natan michvodo le-vasar ve-dam
The Artscroll siddur says we don't do the full blessing (with the name of God) because he's democratically-elected, term-limited, and the like; the Tzitz Eliezer holds similarly.
Rabbi Ovadiah Yosef and the Piksei Teshuvot feel that the president is powerful enough to warrant the bracha; however, Nivchar MiKesef OC3 holds that no matter how powerful, if the ruler is going out as a commoner, you can't say the bracha as where's the royal pomp and circumstance? Therefore if the President is visiting a foreign country, "just wearing a suit like any commoner", with a non-showy security detail, Rabbi Yosef advises playing it safe and saying it without the name of God.
However, that same responsum of the Nivchar MiKesef that requires some pomp and circumstance also implies that seeing the royal residence meets that condition. So if you extended Rabbi Yosef's line of reasoning, you would probably conclude that meeting the President at the White House would get the full bracha with the name of God.
Practically, though, between the questions of both pomp and power, it seems the common practice is to say it without the name of God.