If my Monk uses an action to cast Blade Ward, can I use my bonus action to make an unarmed strike?
I suppose you are referring to the Martial Arts feature:
When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff, you can also make an unarmed strike as a bonus action, assuming you haven't already taken a bonus action this turn.
Using Blade Ward is not using the Attack action (it’s Cast A Spell), so, no, you can't use your bonus action to make one unarmed strike (punch).
The general ruling for using bonus actions to make attacks (Two-Weapon Fighting) also states:
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Again, using Blade Ward is not taking an Attack action.
As neither the general ruling or the martial arts feature apply, it would only be possible if the cantrip itself says "You can then use your bonus action to make an unarmed strike". Blade Ward doesn't.
Just for completeness sake, I would like to clarify that this answer does not change even if the spell itself makes an attack roll, such as Shocking Grasp does. The action you are taking is still Cast a Spell, not an Attack action.