The base weapon attack that's part of the cantrip doesn't become magical.
Rules designer Jeremy Crawford unofficially addresses this question in a 2017 tweet:
Is the attack made as part of Booming Blade magical? It's delivered as part of a spell, but has its "usual effects" and is a prerequisite for the spell to work.
The booming blade spell isn't intended to make the required weapon attack magical.
So the weapon attack you make as part of the spell is not magical on its own. The two cantrips call for a weapon attack made as part of the cantrip, but the weapon attack on its own is not stated as being modified to become magical; there are simply additional effects to the attack that are granted by the cantrip.
...But the extra damage is magical.
The booming blade spell description says (SCAG, p. 142; emphasis mine):
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.
Similarly, green-flame blade says (SCAG, p. 143; emphasis mine):
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and green fire leaps from the target to a different creature of your choice that you can see within 5 feet of it. The second creature takes fire damage equal to your spellcasting ability modifier.
When the caster of either cantrip reaches higher levels, the initial attack does extra damage as well (in addition to boosting the secondary damage).
Per Crawford's unofficial guidance above, the required weapon attack remains nonmagical; the fact that the spell calls for a normal weapon attack doesn't change the properties of the weapon. However, the added damage done by the spell itself is magical, as it's caused directly by the spell. This is true both of the secondary damage of both spells, and of the extra initial damage both spells do at higher levels.
