There aren't a lot of absolute pros and cons to this argument, so the answer is 'it depends.' Some scenarios with different conditions that affect this decision might be:
Client-server app
One example of a place where it might be appropriate to do this is an older 4GL or rich client application where all database operations were done through stored procedure based update, insert, delete sprocs. In this case the gist of the architecture was to have the sprocs act as the main interface for the database and all business logic relating to particular entities lived in the one place.
This type of architecture is somewhat unfashionable these days but at one point it was considered to be the best way to do it. Many VB, Oracle Forms, Informix 4GL and other client-server apps of the era were done like this and it actually works fairly well.
It's not without its drawbacks, however - SQL is not particularly good at abstraction, so it's quite easy to wind up with fairly obtuse SQL code that presents a maintenance issue through being hard to understand and not as modular as one might like.
Is it still relevant today? Quite often a rich client is the right platform for an application and there's certainly plenty of new development going on with Winforms and Swing. We do have good open-source ORMs today where a 1995 vintage Oracle Forms app might not have had the option of using this type of technology. However, the decision to use an ORM is certainly not a black and white one - Fowler's Patterns of Enterprise Application Architecture does quite a good job of running through a range of data access strategies and discussing their relative merits.
Three tier app with rich object model
This type of app takes the opposite approach, and places all of the business logic in the middle tier model object layer with a relatively thin database layer (or perhaps an off-the-shelf mechanism like an ORM). In this case you are attempting to place all the application logic in the middle-tier. The data access layer has relatively little intelligence, except perhaps for a handful of stored procedured needed to get around limits of an ORM.
In this case, SQL based business logic is kept to a minimum as the main repository of application logic is the middle-tier.
Overhight batch processes
If you have to do a periodic run to pick out records that match some complex criteria and do something with them it may be appropriate to implement this as a stored procedure. For something that may have to go over a significant portion of a decent sized database a sproc based approch is probably going to be the only reasonably performant way to do this sort of thing.
In this case SQL may well be the appropriate way to do this, although traditional 3GLs (particularly COBOL) were designed specifically for this type of processing. In really high volume environments (particularly mainframes) doing this type of processing with flat or VSAM files outside a database may be the fastest way to do it. In addition, some jobs may be inherently record-oriented and procedural, or may be much more transparent and maintanable if implemented in this way.
To paraphrase Ed Post, 'you can write COBOL in any language' - although you might not want to. If you want to keep it in the database, use SQL, but it's certainly not the only game in town.
Reporting
The nature of reporting tools tends to dictate the means of encoding business logic. Most are designed to work with SQL based data sources so the nature of the tool forces the choice on you.
Other domains
Some applications like ETL processing may be a good fit for SQL. ETL tools start to get unwiedly if the transformation gets too complex, so you may want to go for a stored procedure based architecture. Mixing Queries and transformations across extraction, ETL processing and stored-proc based processing can lead to a transformation process that is hard to test and troubleshoot.
Where you have a significant portion of your logic in sprocs it may be better to put all of the logic in this as it gives you a relatively homogeneous and modular code base. In fact I have it on fairly good authority that around half of all data warehouse projects in the banking and insurance sectors are done this way as an explicit design decision - for precisely this reason.
