These programs tend to flag all interactions, making it difficult for the patient to interpret the reaction's significance.
A decreased effect is usually not as dangerous (or as noticeable) as an increased effect (depending on the medication, of course), for example decreasing the metabolism of coumadin - a blood thinner - or it's active metabolites by adding an agent which inhibits it's clearance (e.g. an antifungal agent, ketaconazole). This can result in a life-threatening increased effect of coumadin.
Note also that not all drug checkers are alike in their ability to calculate risks/side effects.
For example, at Web.MD, the combination of ketoconazole oral + warfarin oral will give you a "serious" risk:
ketoconazole oral will increase the level or effect of warfarin oral by altering drug metabolism
But Drugs.com gives only a "moderate" risk, with the "professional" option going into more detail:
MONITOR: Azole antifungal agents that are potent inhibitors of CYP450 3A4 such as itraconazole, ketoconazole, and posaconazole may increase the plasma concentrations and hypoprothrombinemic effect of warfarin... (etc.)
MANAGEMENT: Patients receiving warfarin should be closely monitored during concomitant therapy with azole antifungal agents that are potent inhibitors of CYP450 3A4. The INR should be checked frequently and warfarin dosage adjusted accordingly, particularly following initiation, change of dosage, or discontinuation of azole antifungal therapy. The same precaution may be applicable during therapy with other coumarin anticoagulants, although clinical data are lacking. Patients should be advised to promptly report any signs of bleeding to their physician, including pain, swelling, headache, dizziness, weakness, prolonged bleeding from cuts, increased menstrual flow, vaginal bleeding, nosebleeds, bleeding of gums from brushing, unusual bleeding or bruising, red or brown urine, or red or black stools.
And gives you references!
This is not a recommendation of one over another, but an example of how online checkers differ. A third option is to Google only the drugs you're mixing, which will give you not only several calculators but specific papers.
In my opinion, the best advice you can get is from a clinical pharmacologist, or from programs written for professionals by clinical pharmacologists. These programs cost money.
You might be wise to approach drug interactions by asking both your doctor (when the drug is prescribed) and your pharmacist; if they have different advice, they should resolve it by discussing with a clinical pharmacologist.
Even though I check for and advise patients about possible drug interactions, I always tell them to check with their pharmacist as well (in fact, it's a written part of my discharge instructions when I've prescribed a drug, mostly because many patients don't report all the drugs they are taking, don't consider OTC medications as drugs, etc.) This is a case where two heads are better than one.