Effects like Cutting Words (PHB, p. 54) and Spirit Shield (XGtE, p. 10) reduce damage by a roll, irrespective of damage type (PHB 196). However, single damage rolls can have multiple damage types, as confirmed (for wording) by Jeremy Crawford in this tweet.
How do you know which part of the damage is reduced?
This could actually make a difference to the amount of damage taken if, for example, the creature taking damage is resistant to one or more of the damage types involved.
Cutting Words: … When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature's roll.
Using this ability to expand upon the examples found in this related question and this other related answer:
Barny, an Ice Devil (MM, p. 75) hits Bobby, rolling 9 slashing damage and 11 cold damage.
Bobby's ally, Vera, is a level 10 bard of the college of Lore, and uses her Cutting Words ability on the attack's damage roll. She rolls a 9, which is subtracted from the Barny's damage roll.
Bobby is wearing his Boots of the Winterlands (DMG, p. 156), giving him resistance to cold.
We know that "resistance and then vulnerability are applied after all other modifiers to damage" (PHB, p. 197) and both slashing damage and cold damage are treated as "one big damage roll" (see the JC tweet), so is all the 9 slashing damage removed by cutting words, and then the cold damage halved for 5 total damage, or is 9 of the cold damage removed by cutting words, leaving 2 to be halved for a total of 1 + 9 = 10 damage taken by Bobby?
Who makes this decision?
When an effect (like cutting words) reduces damage without specifying type from a source containing multiple damage types, what part of the damage is reduced?
Best Answer
The Bard decides, but it is somewhat table dependent
As you've quoted for cutting words, the mechanic states:
When damage is being rolled, each type is rolled separately: different die, different modifiers, different functional rolls (but still counts as One Big Roll with regard to damage.)
The bard is seeing the damage coming in and using their mechanic to reduce the damage they want to reduce.
Other tables may differ
However, there is no written rule on this and a DM could easily say it was their decision. But I'd suggest to let the Bard bard and choose when and how they want to apply their mechanic.
But you have to pick one from somewhere in some circumstances (but not all)
Due to issues regarding resistance, vulnerability, there needs to be a mechanic where you choose which damage type to Cut. If there are none of the mechanics in play, then it doesn't matter - but if there are, you need to separate out the types otherwise you have no way to determine how and where to cut the damage.
Crossover damage
In the cases where the damage reduction is greater than any single damage type's value, then the remaining reduction would cross over to the other damage type. The bard has targeted Damage Type A, and the remainder (if any) goes to Damage Type B.