I got some numbers tallied regarding losing streaks and their frequencies in Word Series participants.
There's only ever been one team to have a double digit losing streak, 11 games, and reach the World Series. That was the 1951 Giants.
The 1953 New York Yankees lost 9 games in a row during the season and went on to the Series.
There have been 6 teams that have had 8-game losing streaks and reached the World Series: 1922 Yankees, 1961 Reds, 1980 Royals, 1990 Reds, 2006 Cardinals, and 2007 Rockies. Of those 6, only the 1990 Reds and the 2006 Cardinals won the World Series.
12 teams had 7-game losing streaks and made the World Series. So it's infrequent, but possible. 43 teams had 6-game losing streaks and made it. 57 teams had 5-game losing streaks. The most frequently occurring maximum losing streak was 4 games -- 82 of the teams in the World Series had losing streaks of at most 4 games.
A small number had seasons where they never suffered more than 3 defeats in a row, 16 times. Obviously, they were all very good teams. 1957 was the only year in which 2 of those 16 teams met in the World Series. Both the Milwaukee Braves and New York Yankees had never lost more than 3 games in a row that season. Teams that have at most 3-game losing streaks have been getting less frequent over time. The first 8 of them occurred between 1903 and 1932, 30 years, we've seen only 8 more over the last 77 seasons. It's probably because over time measures have been taken to assure more equitable talent distribution along with all teams, even the bad ones, becoming better and more efficient at evaluating talent and needs. So the good teams can't rise as far above their peers as they could in the earlier years of baseball.
The 1989 San Francisco Giants, who got swept in the World Series 4 games to none, were the last team to have been in the World Series and had at most a losing streak of 3 games in the regular season. They had 6 3-game losing streaks during the regular season.