Please note, as of February 28, 2019, Curbi is no longer available and some of the suggestions below are outdated.
Please see Apple Screen Time a Huge Win for Everyone, Including Parents! for a great solution to use instead.
As parents, we certainly can’t be with our children constantly to monitor all the inappropriate material they may encounter when using a smart phone. We we can help them learn to make good choices and discuss what to do when they do come across inappropriate content. We can also use technology tools to help limit the amount of exposure to inappropriate content which I highlight below:
One of the quickest, easiest ways to put a filter on a cell phone is to use Restrictions (more details and instructions). On Apple and some Droid devices, Restrictions allow you to create a separate password from your child's screen lock and limit web content (on iPod/iPad/iPhone), app installs, purchases, ratings for movies/apps/explicit language, and much more. I am more familiar with Apple than Droids, but from what I've read it looks like Droid Parental Controls/Restrictions unfortunately won't filter web content, so see option B farther below.
The default setting for websites that are restricted through Apple iOS is pretty strict, but seems to offer the best option at this time. You can add a few sites to a whitelist, but your child may need to go to a desktop computer with a more robust filter or need you to enter in your Restrictions password to temporarily access some sites. I think this inconvenience is a small price to pay for the benefits of having this safeguard in place.
[Added April 2015]
Curbi is a great option for parents to filter and monitor their child's Droid or iPhone/iPad and provides valuable data to spark conversations around the use of technology. Read more here in a later post.
There are also services you can purchase that will help you filter your child’s phone such as SafeEyes, MobiCip and NetNanny, but it is important to realize that these products only filter the internet when accessed through that app. These products have you turn off the cellphone's web browser, such as Safari on an iPhone and Chrome on a Droid, and then they have you also turn off the ability for your child to install and remove apps without your password. This is their solution to preventing your child from not using the filtered web browser/app. (For example, see NetNanny's FAQ Disclaimer #9: Can Net Nanny filter inappropriate content within "other" apps?) However, if your child accesses a webpage link through any other app with its own built in web browser such as Schoology, Instagram or Facebook, it bypasses the filtered web browsing app. This is a major limitation, which is why I feel that cellular providers need to offer a solution to filter all internet traffic to and from the device, which leads us to Option C. But don't get optimistic about this third option:
Some cell phone providers may offer content filtering services parents can choose to activate. My family's carrier, Verizon Wireless, used to do this for 3G services, but hasn't for 4G LTE now for years. So there is no option for parents to choose this for a smartphone, despite the misleading promise that Verizon lists on their website stating that these services are available to parents.
From looking around, it seems that AT&T, Sprint, and TMobile also don't offer these services for smartphones, either. If you know otherwise, please let me know. To learn more, you can Google your service provider and add the words "parental controls", but be sure to read the fine print to see if you can filter websites your child can access through her/his smartphone. I had started a petition to get cellular companies to offer these services for 4G LTE but that closed.
Some phone companies like Kajeet specifically offer filtered phones for children with a variety of additional control options. Kajeet does offer filtering for Sprint devices, but that won't cover WiFi, so that brings you back to Option A and A2. They also offer unlocked and contract free phones, which unfortunately are a no go for most parents who want everything on one bill under their Family Plan, etc.
Other filters to consider: