It's an entry level DSLR, but used properly will be capable of excellent results. It's really about understanding the lighting and composition that makes a good photograph, and then knowing how to operate your camera to capture what you see.
One thing that might be worth considering, if you're planning to take your camera out-and-about with your child, is whether a more compact model may be worthwhile (such as a micro-four-thirds mirrorless, or a good compact like Sony's RX100 series)? Even though the D3300 is quite small by DSLR standards, you might still find that it's inconvenient to carry around with you on trips out if you're having to also carry all the rest of the baby stuff. A more compact camera can be easily dropped into a bag or even a pocket, and you're much more likely to take it with you and then be able to capture those moments. Mirrorless cameras have all the flexibility of their larger DSLR cousins, but without (in most cases) the bulk. Compacts have less flexibility, but the good ones can produce fantastic photos.
Otherwise though, the D3300 is by all accounts a great camera. I have a D3200, the preceding model, and have been extremely pleased with it and the results I've achieved.