And although coconut oil can be antimicrobial, you always have to consider the possibility that something with positive effects could have unintended side effects in some people. One animal study suggested that lauric acid could have harmful effects on autoimmunity, although the study may not have been that applicable to real life humans. Possible immune modulating effects of different fatty acids in coconut oil are largely unknown, so this is where dose really matters: a tablespoon a day is unlikely to have an impact, but high doses (especially in the contexting of having an existing immune condition) could be playing with fire.
According to some studies, medium-chain fats offer better protection from infections than longer-chain fatty acids do. A study published in The Journal of Nutritional Biochemistry found that fatty acids and monoglycerides with chain lengths varying from 8–12 carbons were found to be more strongly antiviral and antibacterial when added to milk and formula than long-chain monoglycerides. Medium-chain lipids added to milk and formula inactivated a number of pathogens including respiratory syncytial virus (RSV), herpes simplex virus type 1 (HSV-1), haemophilus influenzae and streptococcus.
Even then, though, I don't really recommend deliberately trying to increase your saturated fat consumption. Animal studies show that coconut oil by itself will push cholesterol (and possibly triglyceride) levels much higher than compared to fish oil.  Dr. Weil gives a very balanced response for those interested.  Please read these links on The Potential Dangers of Saturated Fat and The Potential Dangers of the Atkins/Paleolithic/Low Carb Diet for more information.