I would like to record some songs, especially with a Christmas theme. I will soon purchase some recording equipment and I am wondering how I can reduce the breath sounds.
The main thing is probably not to put the microphone too close to the fipple, or perhaps I should say too close to your mouth. Note the placement of the mic in this video:
That mic looks like a Shure Beta 87, which is far from expensive in microphone terms. If it's good enough for Mary Bergin, it's probably good enough for the rest of us.
If you want to completely eliminate the breath sounds the best way is to record two as-near-as-possible identical takes of the tune only taking breaths in different places; then edit the two together to make one breath-free take. It makes an interesting and slightly un-natural sound so should be used sparingly, as when you start recording whistle or flute it's quite surprising how much the breath points affect the pulse and feel of the music.
If you want to reduce the sound, try taking shallower breaths more frequently, and/or work on your diaphragm technique so you can play longer phrases, and generally try to ensure that your breaths are as quiet as possible (singers' throat lozenges can help clear the airways for example). For recording you could also go back through and use the recording software to manually duck the volume of your breaths, but that's laborious and also addressing the consequence of the problem rather than the cause.
If you're gasping or gulping air noisily, you're almost certainly doing it wrong and can work on that; but a certain amount of breath noise enhances the 'live' sound of the instrument.
I can't answer your question directly since I've never recorded a penny whistle. However, I CAN answer in more general terms.
The first approach would be utilizing proper recording technique. Ideally, you should be able to place the mic reasonably to your sound-source (i.e. the whistle's end... not sure what the term is) while also pointing the microphone partially (or even better, COMPLETELY away) from the source or sources that you don't want to hear (i.e. your lips / mouth). This approach is especially effective when using microphones with strongly directional polar patterns. Read more about polar patterns here.
The second approach would be directly editing your recording. By splitting the sections with egregiously loud breaths, reducing the volume and cross-fading them into the original audio you can reduce unwanted breath sounds. How feasible this is, depends on how often the breath sounds occur and how much the breath sounds bleed into sounds that you actually want. So while this works for a lot of vocal recordings, it might not work for your purposes. You'll have to make that determination yourself.
The third approach would be through the mixing process. Using volume automation, de-essing and/or eq you can chop out offending sounds and frequencies while, hopefully, keeping the bread-and-butter of recording intact. This is a whole topic by itself and can't be adequately covered here, but for more info on eq and common frequency ranges, you can check this article.
There is no clear cut answer to your question unfortunately, but by combining a little creativity and some of these approaches you should be able to yield a better result. Remember that with recording, the more you do it, the better your ears will develop and the more your instincts can take over for these types of situations.
If you are recording to the computer, then you can also edit the breaths manually on the timeline. This is a little more time consuming, but it produces the best results in my opinion. This kind of editing is also commonly done with vocalists, who sing directly into the microphone and are usually fairly close to the mic. Once you get the hang of it the editing can actually be done fairly quickly. Most recording software allows you to split the audio and adjust the volume of an individual clip without affecting the others. By making two splits on either side of the breath you can reduce the volume of the breath or remove the breath entirely.
If you decide to do it this way, then you can place your microphone wherever you want to get the best tone of the whistle without needing to worry about minimizing the breath sounds as you will be controlling those after you record.