As MMS(Multi-Media-Messaging) is one of multiple ways this exploit can be run, you can prevent this from MMS exploits. Stagefright on its own is not an exploit. Stagefright is a multimedia library built into the Android framework.
The exploit was found in a multimedia tool hidden at deep system
level, so almost all Android variants that have the tool at their core
could be easily targeted. According to Zimperium zLabs’ study, a
single multimedia text has the ability to open up your device’s camera
and begin recording video or audio, and also give hackers access to
all your photos or Bluetooth. Fixing Stagefright would require a
complete system update (which still hasn’t been reported to be
released by any OEM), as the exploit is embedded to a system-wide
tool. Fortunately, SMS app developers have already took the issue into
their own hands and released temporary fixes to avoid Stagefright to
gain automatic access to your device’s camera by stopping video MMS
messages from running as they arrive. [Source - Android Headlines]
You could use Textra SMS or Chomp SMS from the Play Store which claims that it is capable of limiting this Stagefirght exploit. Both apps Textra and Chomp SMS developed by Delicious Inc. have both received brand new updates that limit how video MMS messages run as soon as they are received by your device.
From the Textra Knowledge Base article,
The stagefright exploit can occur when any SMS / MMS app creates the
MMS video thumbnail that it shows in the conversation bubble or
notification or if a user presses the play button on the video or
saves to Gallery.
We have provided a solution for 'StageFright' in Release 3.1 of Textra
out now.
Very Important: In other SMS / MMS apps, turning off auto-retrieve is
NOT enough as once you tap 'download' the exploit potentially becomes active. Additionally you would not get any MMS pics or group
messages. Not a good solution.
According to the developer of both the apps,
the risk of your device being targeted to this new exploit is greatly
decreased by denying MMS messages the ability to automatically run.
How can I protect from Stagefright using Textra?
Turn on the Stagefright Protection
under your Textra app Settings.

Screenshot (Click to enlarge image)
So here is what happens if you activate the app's Stagefright Protection and if you receive a Stagefright exploit message,
- Stagefright Protected: As you can see below, the message was not downloaded and the thumbnail hasn’t been resolved, so if this video has an exploit targeting Stagefright then it will not yet be able to execute its code. The message has a nice “Stagefright Protection” label beneath it.

Screenshot (Click to enlarge image)
- What happens if I click the Stagefright Protected message? : When you hit the Play button on the MMS message: an even bigger box, with an even bigger play button, and an even bigger “Stagefright” label.

Screenshot (Click to enlarge image)
- Do you still want to open the Media and get affected? : Finally, clicking the Play button one last time will bring up a nice warning message reminding you that downloaded videos may contain an exploit called Stagefright.
(Note: there is no known exploit, and if there were its name wouldn’t be Stagefright as Stagefright is simply the name of the multimedia library that is vulnerable to being exploited).

Screenshot (Click to enlarge image)
Pressing the OKAY
button will then bring up whichever video you were going to view, and that’s it. If said video actually did contain an exploit that targets Stagefright then it would, in fact, execute at this moment.
Source : Phandroid
If you are curious about whether you are already affected and a victim of Stagefright exploit then download this app Stagefright Detector from PlayStore which was released by zLabs(Zimperium research labs) which first reported the issue to Google.
Updated: [18-09-2k15]
Motorola has officially released a security patch for Stagefright security issue on August 10 to the carriers for testing and it's release to the public based on the carrier provider. It is mentioned in the forums that,
As soon as a patch is ready you will see a notification on your phone
to download and install the update. We encourage everyone to
periodically check if they have the latest software by checking in
Settings>About Phone>System Updates.
And if you using Motorola and still din't get the security patch you can read the following to avoid the security risk of under attack,
What can I do to protect myself if my phone does not have the patch?
First, only download multimedia content (such as attachments
or anything that needs to be decoded to view it) from people you know
and trust. You can disable your phone’s capability to download MMS
automatically. That way you can only choose to download from trusted
sources.
- Messaging: go to Settings. Uncheck “Auto-retrieve MMS.”
- Hangouts (if enabled for SMS; if greyed-out, no need to take action): go to Settings > SMS. Uncheck auto retrieve MMS.
- Verizon Message+: go to Settings > Advanced settings. Uncheck Auto-retrieve. Uncheck “Enable weblink preview.”
- Whatsapp Messenger: go to Settings > Chat settings > Media auto-download. Disable all video auto downloads under “When using
mobile data,” “When connected on Wi-Fi” and “When roaming.”
- Handcent Next SMS: go to settings>Receive message settings. Disable auto retrieve.
Read more on :
How to protect from stagefright vulnerablility?
What are the other attack vectors for Stagefright?