The Dodgers are the six-time defending National League West champions, while the Brewers needed a tiebreaker to win just their fourth division title in nearly five decades of existence.
None of that really matters when it comes to who will win the upcoming NL Championship Series, but it does go to show you just how differently the recent history of these two teams has been. While the Dodgers are in the midst of long-running success, the Brewers were a fourth-place team as recently as 2015 and '16, and they saw their turnaround fueled by a multitude of successful trades and signings.
Milwaukee has also won 11 straight games, dating back to Sept. 22. The Brewers have baseball's hottest hitter in right field, a deep bullpen that more than makes up for any weakness in the rotation, and a manager who isn't afraid to deploy his pitchers aggressively. They may not have the postseason experience of the Dodgers, but then again, that doesn't really matter, does it?

Let's go position by position to see who has the edge. 
Catcher
Yasmani Grandal (.241/.349/.466, with 24 homers) had the third-best hitting line of the 27 catchers who had 300 plate appearances, and he also rated as baseball's best pitch framer (+17 runs), as he so often does; there's a strong argument to make that only J.T. Realmuto had a better all-around year among regular catchers. Despite the unexpectedly strong October run of Erik Kratz (.236/.280/.355 during the season) and the strong defense of Manny Pina (.252/.307/.395), neither Brewers backstop can match up to Grandal. 
Advantage: Dodgers

First Base
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The Brewers have a 28-year-old breakout star taking advantage of an opportunity to play after years in the Minors. But so do the Dodgers, almost exactly; both Jesus Aguilar (.274/.352/.539, with 35 home runs) and Player Page for Max Muncy (.263/.391/.582, with 35 home runs) were born in the summer of 1990. So, why do the Dodgers get the edge here? Because Muncy has a 39-point edge in OBP and a 43-point edge in slugging, and because Los Angeles also has David Freese, who's made a career out of smashing lefty pitching, and Milwaukee has a pair of southpaw starting pitchers.
Advantage: Dodgers
Second Base
Speaking of similarities: Both teams have starting second basemen who spent much of the year playing other spots, and both teams acquired veteran sluggers in July who failed to add much value to their new clubs. (James Dozier hit .182/.300/.350 for the Dodgers after arriving from Minnesota; Jonathan Schoop hit .202/.246/.331 for the Brewers after coming over from the Orioles.)
As for those starters, however, Travis Shaw (.241/.345/.480, with 32 homers) hit well and fielded capably after a mid-season shift from third to second, and somehow he still played more second-base innings than Enrique Hernandez (.256/.336/.470 with 21 homers) did. Shaw gets the small edge due to a better track record with the bat, though he may be somewhat vulnerable to a lefty-heavy Dodgers rotation after hitting just two of his 32 homers against lefties in 2018.
Small advantage: Brewers

Shortstop
We should note that Orlando Arcia has had some big moments lately, including going 4-for-4 in the NL Central tiebreaker, but his .236/.268/.307 line made him one of the weakest hitters in baseball this year. Hernan Perez, who is expected to be in the lineup vs. Dodgers starter Clayton Kershaw in Game 1, had a .253/.290/.386 batting line on the season, but he did post a .783 OPS in 125 plate appearaces vs. lefties this year. Of course, Manny Machado (.297/.367/.538, with 37 homers) put up his usual big offensive season, even if Arcia gives the Brewers a clear defensive edge when he's in the lineup.
Big advantage: Dodgers

Third Base
Mike Moustakas (.251/.315/.459) provided the Brewers with a league-average bat and average fielding, which is more valuable than it sounds, because it allowed Shaw to move to second to replace the ineffective Jonathan Villar. Of course, Justin Turner (.312/.406/.518) is a full half-decade into being one of the most dangerous hitters in the game.
Advantage: Dodgers

Left Field
With Chris Taylor no longer playing nearly as much infield as he did earlier in the year, he sees most of his time here, platooning with Joc Pederson. Taylor (.254/.331/.444) hasn't had a strong season, but Pederson (.248/.321/.522) has, putting up a top-25 slugging rate among players with as many plate appearances as he had. And since none of the three are strong defenders, that slugging edge gives the Dodgers a small advantage over Ryan Braun (.254/.313/.469).
Small advantage: Dodgers
Center Field
Though the Dodgers cycle Taylor, Hernandez, and Pederson through this spot as well, this has largely been Cody Bellinger's job, and he (.260/.343/.470) has hit basically as well as Lorenzo Cain (.308/.395/.417) has, with Bellinger having the power edge and Cain having a large on-base advantage. The difference here is that while Bellinger plays a solid outfield (+4 Outs Above Average), Cain (+18 OAA) is one of the best defenders in the sport. That gives Milwaukee a solid edge. 
Advantage: Brewers

Right Field
Yasiel Puig (.267/.327/.494) and Matt Kemp (.290/.338/.481) each had solid seasons for the Dodgers, with batting seasons that were about 20 percent above-average. Of course, Christian Yelich (.326/.402/.598, with 36 home runs and 22 steals) is the slam-dunk NL Most Valuable Player Award winner, so no, this one's not close. 
Big advantage: Brewers

Starting Pitchers
For all their similarities, these two teams could not look more different on the mound. The Dodgers are going to roll out a strong quartet of Kershaw, Hyun-Jin Ryu, Walker Buehler and Rich Hill, who combined for a 2.82 ERA this year; they have so many starters that they've sent Thomas Stripling, Kenta Maeda and Alex Wood to reinforce the bullpen. While Milwaukee's Jhoulys Chacin is better than you think, he wouldn't crack the Los Angeles rotation -- nor would Wade Miley or Giovany Gonzalez, who is starting Game 1 against Kershaw -- and they'll possibly just go with a bullpen day at least once.

That makes this a big edge for the Dodgers, but that's OK, because ... 
Huge advantage: Dodgers
Relief Pitchers
... the Brewers have an equally large -- if not larger -- advantage in relief. It's how their pitching staff is built, after all. If you take what we'll arbitrarily call their top five relievers, including Josh Hader, Jeremy Jeffress, Corey Knebel, Brandon Woodruff and Corbin Burnes, that group had a 2.54 ERA and more than 12 strikeouts per nine innings.  
It's not that the Dodgers' bullpen is bad, necessarily. They actually had baseball's second-best relief ERA (behind Milwaukee, of course) in September, because there's a lot of talent there in names like Caleb Ferguson, Ryan Madson, Pedro Baez and Scott Alexander, in addition to the trio of former starters. That said, Kenley Jansen's season has been something of a roller coaster, and even if it wasn't, this group hasn't shown the same dominance as the Milwaukee relief crew.
Huge advantage: Brewers

Prediction
This one is close, maybe closer than you might think, and could easily go either way. We'll take the Dodgers in 7.