I'm making a rogue(14)/monk(6) character. In DnD Beyond, the saving throws that this character is proficient in only reflect the rogue class. So if you multiclass, what is the rule for saving throw proficiencies? Rogue is proficient in Dex/Int saving throws and Monk is proficient in Str/Dex, so multiclass Rogue/Monk should be proficient in the Str/Dex/Int save throws?

From the multiclassing rules:

When you gain your first level in a class other than your initial class, you gain only some of new class’s starting proficiencies, as shown in the Multiclassing Proficiencies table.

And saving throws is not in the list. So you only get the saving throw proficiency from your initial class.

What Thezzaruz said. Make sure you read those multiclassing rules so you know how that works (DDB does a good job, but not 100%, at following them automatically). It also affects weapons, armor, skills, and tools, so check that chart.

Notably, additional save proficiencies from a subclass or later class feature (like monk's level 14 diamond soul) still apply. Only level 1 proficiencies are limited.

One of the main reasons people use to pick which class they take first is to decide which set of saves they want.  Each class has one of the common saves (Wis, Dex, Con) and one of the uncommon (Str, Int, Char) that they are proficient in.  

But, you should also note that when a rogue hits level 15, they get Wisdom saves.  How set are you on Monk 6?

Yeah you need to pick either Str/Dex by starting Monk at 1st level or Dex/Int by picking Rogue at 1st level.

Which is better may depend on the setting/campaign you're in. Str saves aren't super common but could be useful from time to time not getting knocked around by control/knockback effects. And Int is also not super common unless you're going up against a lot of psychic enemies, a lot of those mind-targeting psychic damage effects can target Int saves.

You can also use the Resilient feat to get that extra saving throw proficiency, if you have room for it.