A barbarian, while raging, gains a damage bonus when striking with a "melee weapon that uses Strength".
He can also gain advantage to his attacks for the turn when he chooses to attack recklessly with a "melee weapon that uses Strength".
A monk, while using a monk weapon, can choose to use Dexterity instead of Strength for his attack & damage bonus.
Technically, most melee monk weapons (such as a quarterstaff) use Strength by default, so if a monk-barbarian, who wields a Strength-based weapon, decides to use Dexterity instead, can he still benefit from the Reckless Attack feature and the Rage damage bonus ?
Best Answer
No, because the weapon isn't using Strength any more.
Down in the Making An Attack section (PHB p193-194) we get this...
ALL melee weapons use Strength, by default.
The rule for the Finesse tag on p147 says...
This gives you an exception....you may use Dex instead of Str when using a Finesse weapon.
the rule for a Monk says
Emphasis Mine to show that using Dex instead of Str is optional for a Monk.
Notice how similar the phrasing is between the Finesse rule and the Martial Arts rule.
Specific over General
The fact that the general rule for melee weapon attacks is "Use Strength" with the specific exception that "Finesse Weapons break the rule," coupled with the fact that the Barbarian class description has to explicitly say:
despite the fact that all melee weapons use Strength by default makes it pretty clear that, in order for the Rage damage bonus to work, you have to actually use Strength for your attack/damage roll. Otherwise, it would work with Finesse weapons used with Dex, too...and the call-out that it must be a melee weapon that uses Strength is meaningless.