Statistically speaking, there's little difference between the most popular opening moves, i.e. 1.e4, 1.d4, 1.Nf3, 1.c4.
For instance, in the Game Database of ChessTempo on sees the average rating of the players and their average performance:
- 1.e4: average 2417, performance 2455
- 1.d4: average 2436, performance 2474
- 1.Nf3: average 2431, performance 2465
- 1.c4: average 2435, performance 2467
Although these four opening moves are approximately equally good, they can lead to different kind of positions (tactical vs. positional, lots of theory vs. little theory, etc...). Therefore, the opening move (and the opening in general) does have a big influence on the outcome of the game, depending on the players and their playing styles.
When playing against engines, it is probably a good idea to go for closed positions. In these type of positions, the tactical power of engines is less dominant, while strategical insight becomes more important.