Winds do cause ocean waves. Go to the beach on a windy day and you will see waves run farther up the beach than on a calm day. This occurs because higher winds generate higher waves.
To be fair, extreme winds can actually change the heights of the tides. During hurricanes the height of the high tide increases due to lower air pressure above the water. This change is called a storm surge, but it only occurs noticeably during severe storms.
The Moon is not the only cause of the tides.
If the Moon was closer to Earth than it is today, the most important effect on the tides would be higher tides than today.
Remember that the Sun's gravity also has a significant effect on Earth's tides. After all, the Sun is 27 million times more massive than our Moon. However, the greatest distance between the Sun and the Earth -nearly 400 times the distance between the Moon and the Earth - significantly diminished the Sun's tidal effects. Combining the effects of the Sun's greater mass and greater distance leads to the result that the Moon creates tides about twice as high as those created by the Sun. So this value would be less than today if the Moon was closer to Earth.
The sun also produces tides on Earth, so without a moon there would still be ocean tides (2 per day), they would simply be smaller.
Since the moon's gravity is what causes tides, if the moon was closer to the Earth its gravitational influence would be greater so the tides would be higher. Since the twice-a-day tides are due to Earth's rotation through the tidal bulge, we would still have 2 tides per day.
An additional note - since the moon would have to move faster in its orbit if it was closer to Earth, full moon to full moon would be less than the current 29 days.