Disclaimer - I am associated with Dekho, but the following is my personal attempt at answering the Q, and I hope I have not come across as biased
To recommend something, it is all about your requirements.
TBH, we need a better understanding of what your organisation wants - how many users, existing infrastructure/software, in-house skills, etc.
- wants to publish their maps online
To publish maps online, as Stephen says = u need a server back end to serve up the services, and a mapping front-end to consume the services.
The first Q you need to ask is Esri or Non-Esri?
- looking at not spending much money
The back-end
Esri costs money.
You would need to incorporate ArcGIS Server, but you might also want to check out if ArcGIS Online meets your needs. You can do some great things with AGO, especially if you make use of ArcGIS Explorer Online.
Perhaps you are already an existing Esri shop and therefore it might make sense to request an evaluation of ArcGIS Server from your local distributor to see if it fits your needs?
Open Source is an excellent alternative, and MapServer & GeoServer have come along way. One easy argument from Esri is that with Open Source you lack the support and bigger picture (i.e. integrates with rest of ArcGIS products well) that ArcGIS Server provides, but there are outfits that will help implement and support open source implementations. Personally, I think that the back-end geoprocessing from ArcGIS Server excels over the current offerings from OpenSource back-ends (for now).
This post is an excellent roundup of the best open-source back-ends.
The front-end
OpenLayers is an excellent front-end choice if you go down the open-source route, browse their gallery to get an idea of the sorts of front-ends you can develop.
If you have ArcGIS Server in the mix, you can go down one of the APIs route. You have not mentioned how you want to consume your maps, but the good things with Esri, is that they have nearly every front-end covered (iOS, Android, Windows, Javascript, Flex, Silverlight, etc)
So - this then leads to the next question.
Do you build a front end application from the ground up Vs buying into a 'Off the shelf' application such as Dekho?
If you want to build from the ground up, but also do not want to spend much money:
- Do you have the in-house skills to develop this and maintain it?
With web mapping evolving so quickly at the moment, you also need to consider how futureproof will the application be, and what is the expected shelf-life for the app?
Building from scratch does give you full control over the look, feel and functionality of the application - but at a cost.
Some arguments for going down the COTS route (regardless if this is Dekho or something else):

In short - if your requirements are not basic, then development times can shoot up.

COTs are:
- quicker to implement and no development times.
- In theory = reduced risk. Why? Because they have already gone through a period of testing and many other clients have guinea pigged the applications in various environments and helped iron out bugs.
- Supported - If you have a problem with a custom app, you are usually on your own. However, support with Esri relies on you paying maintenance fees.
- You get upgrades with COTs,
(Ignore the community and proven track record, as this can apply to non-COTs and non-Esrias well).
At the time of writing, a very good middle ground between building from the ground up and a COTS product would be the ArcGIS Viewer for Flex/Silverlight These viewers and their core widgets are supported and updated to match the functionality from ArcGIS Server. They also have a good framework to bolt on custom widgets for your specific business needs.
- support for high-res aerial imagery?
Both ArcGIS Server and the Open Source offerings, provide the means to dish out aerial imagery to client applications.
With ArcGIS Server usually a cached service is the best approach, but read here to see if it would be worth looking into Image Server.
Again, cached/image services from ArcGIS Server can be consumed in all of the Esri clients.
You can serve them up as WMS if need be, for other non-Esri apps as well.
That works vice versa - An Open Source back-end can also serve up your imagery via WMS, and then consume this in clients like ArcMap.
Hope some of that helps. I have only dabbled with Open-Source, and therefore cannot provide a detailed answer from an organisational perspective - I will let someone else pitch in with an answer that has more focus on the pros/cons of going down the Open Source route.
Let me know via comments if you require additional info.