There are several alternatives you have, let's talk about reducing encounters first:
Budgeted XP and encounters
A good way to adjust the difficulty of an encounter is to look at the XP budgeted for an encounter as expected by the published module and adjust it to the size of your group. Guidelines for this can be found on page 82 of the DMG or page 56 of the WoTC DM Basic Rules. This involves calculating for each creature's XP multiplied by the appropriate Encounter Multiplier factor for the number of creatures the group faces, this can be found on the same DMG page.
Page 83 of the DMG also takes into account the size of the adventurous part, saying that it increases or decreases the multiplier by .5 according to the part size. In your case it would increase the encounter multiplier by .5 because it only has 3 players and published adventures are supposed to be played with 4.
For example, if the published meeting includes 4 CR 1 creatures with a total value of 800 XP. Looking at the match multiplier table, 4 creatures yield a 2x multiplier then you have a Planned budgeted XP of 1600. Looking at the XP Thresholds table by character level, we see that the encounter is intended to be a Medium difficulty I find for 4 fifth level characters.
But since the size of your party is small, increase your encounter multiplier to 2.5x. So if you don't change the match, you would have a XP budgeted from 2000Which is slightly more of medium difficulty for a group of 3.
So how do we modify the encounter to be medium difficulty for 3 parts? We know that for 3 fifth level characters, a medium difficulty encounter is budgeted at 1500 XP, so we can simply draw 1 creature to reduce the budget to scale, since changing the number of creatures from 4 to 3 reduces the multiplier, we have the result of a budgeted XP for your small group of 1500 XP, perfectly adjusted.
That's all well and good, but what if the encounter is against ONE really strong creature?
Modifying and Creating Monsters
Another way to balance encounters is to create your own using the DMG guidelines on page 273 on Create / Modify a Monster (unfortunately, there doesn't seem to be a counterpart to these rules in the Basic DM Rules). This involves taking a creature's essentials and reducing its other stats to adjust it to match the desired Challenge Rating. Everything is detailed in the DMG, so I won't put it here.
Once you have your creature's CR, find its XP value and adjust it for encounter difficulty, as mentioned above. This is your XP budgeted for the encounter.
Let's say, a creature CR 7 with a value of 2900XP (with a multiplier of 1x for 4 characters), would be close to a difficult encounter but much more than mortal for a group of 3. To make it just difficult, the budgeted XP should be 2250 XP, which means the creature must be around CR 6 (5 if your player's characters are lower level). This is where the guidelines for Create a Monster come in, you must modify the creature's existing stats to knock down a pair of CR. This method is fine but cumbersome, unless you can find a resource tool somewhere on the net.
Alternatively, instead of scaling down monsters, you can try scaling up players
1) Give your players hero points
Hero Points is an optional rule found on page 264 of the DMG, it works similarly to Bardic's inspiration and increases the player's chance of success with a direct addition to any roll they make. This is a great way to boost your players and make them feel strong and important. However, players may feel like they are being spoon-fed and don't like that, so it would be better to talk to them about adding this option first, especially if there is a Bard at the party.
2) Give them magic items
Magic items don't fit into encounter difficulty calculations unlike levels and challenge rating, but the more +1 weapons the small game has, the easier encounters will be. There are no guidelines on how much this affects difficulty, but personally, after running several high magic campaigns for a normal size 6th level group that easily has two combat magic items each, you can expect to increase your difficulty by one step. (That is, medium encounters are easy, difficult is medium, deadly is difficult).
Warning, however, you may want to avoid giving them + x Armor, as this can break the game with the PC's super high AC. Instead, consider Adamantine armor (or a similar magic item for light armor or unarmored PC) that denies critics hitting them, denying random casualties. By the way, if you are designing a solo session, this is one of the best elements to give them.
3) Add an NPC adventurer
You could simply add another adventurer of the same level as the party you control. This is an added burden on the DM and affects the player agency a bit, but if you keep the NPC's influence to a minimum, players CAN allow this to happen. The extra muscle will certainly help, but just like the Hero Points option, talk to your players first.
4) Allow characters to accept learners / followers
Another way, related to the above method, is to allow your players to take on a top tier character to the party that they control as learners or followers. This allows a certain degree of help during combat and gives players an alternate character in case the top tier dies. Also, seeing a player talk (and even argue) with himself is golden (why should GMs be the only ones to have to do this?)