That depends on what sort of yarn you use and how big you want the booties to be (I'm going to knit this pattern myself now
If you use sock yarn like the pattern suggests, you can probably use either sizes 1, 2, or 3. Which size needle you want to use will depend on if you want to make it newborn sized or a little bigger.
If you don't use sock yarn, check the label of the yarn; it usually has a little symbol with crossed needles which will say which size needles are recommended. Keep in mind the thicker the yarn, the bigger the bootie.
Knit a gauge swatch, which will give you an idea of how large the final product will be, and then decide if you have the right size needle.