Visiting the "tourist" parts of the Big Cave poses no serious safety hazards. As mentioned above, footwear with balance and traction is a good idea, particularly if you're walking down from the natural entrance; the trail is steep in spots and may be wet due to water dripping from the ceiling. If you're suffering from ear problems e.g. blocked Eustachian tubes, you might find it more comfortable to descend into the cave via the natural entrance rather than the elevator, which descends so rapidly that it may cause discomfort due to difficulty in pressure equalization.
When visiting backcountry areas, whether on tours or on your own, make sure you are properly equipped. Many of the "wild" caves require proficiency in vertical technique that is considerably different than that used in rock climbing. The Park Service's recommendations for any particular cave are usually well thought out and helpful. Make sure you have sunscreen and plenty of water for any hiking; it's sunny, and hot, out there.
fees/permits
The basic fee for the main cave is $6, Children 15 and under are free, good for three days. Headset rentals are an extra $3 per person. A Park Pass $50 annually provides continuing access for the holder and immediate family to this park and other units of the National Park Service. The park also holds several free admission days throughout the year. See the Fee and Reservation page for the dates (http://www.nps.gov/cave/p...).
Several of the ranger-led activities see under "Do" require additional fees. Consult the NPS web site below for more information. Backcountry activities other than surface day hikes extending beyond the ranger-led tours require permits.